Confess in Confidence

There are several professions where there has traditionally been a certain level of confidence between the professional and the client. Solicitors, clergy and medical professionals are the most common examples.

Naturally this is a very useful tool in fiction. You can have an exposition by the criminal and put a person in an unimaginably difficult situation, break their code of conduct or allow a criminal to go free. This trope has three main sub-sets:

Priest-Penitent Privilege: Most often invoked with (or by) Christians, this practice exists in many real world religions. There is often scope for conflict between religious and national law too, as it is often not recognised in court; yet, unlike some other privileges it is, for the priest, absolutely inviolable (For Catholic priests, pardon for breaking confidence has to come from the Pope, and even then, one of the normal conditions of absolution may be forbidding the priest to ever hear confessions again. History provides numerous examples of priests going to jail, suffering torture or even choosing death over breaking the confessional seal). See Confessional.
Doctor-Patient Privilege: Less common in cases of crime but occurs in situations where one person's actions put their well-being or the well-being of another at risk.
Attorney-Client Privilege: Keeping the secret that your client is guilty is often used to show that an attorney is Evil, even if he is just doing what the law requires of him. This is often the strongest from a narrative point of view as the others are often not recognised by the courts but lawyers are frequently required to remain silent.

There is also Spousal Privilege, in that a spouse can not be forced to testify about private conversations.

Often a case of Did Not Do the Research when it's a profession that doesn't have this kind of privilege (librarians, for example, have patron confidentiality as a professional standard but don't have any sort of legal obligation or protection in this matter) unless it's played for laughs.

Similarly to librarians, there is generally no attorney - client privilege for scientists, engineers, architects, etc., who provide data, designs, etc., for pay, but there is usually an contractual obligation to keep the data confidential.

Engineering ethics (which varies from one professional organization to another) generally dictate that if it becomes obvious that continuing the project will result in either violation of the law or unacceptable risk, the engineer must halt work and contact the employer. Only if the employer is not cooperative, the engineer should resign and contact authorities, though not necessarily in that order.

The media do not enjoy legal protection in this manner and may be forced to reveal their sources (on penalty of contempt of court) despite numerous attempts to gain immunity; in what may be an internal case of Not Doing the Research, reporters have been known to claim that they have this right, both in Real Life and in fiction.

Examples of Confess in Confidence include:

Priest Examples

Anime & Manga

Happens in Osamu Tezuka's MW, where Michio taunts Garai by confessing to many of his more horrible misdeeds in church.

Film

Subverted in For Your Eyes Only: After Kristatos has taken the ATAC from Bond, Bond goes into a confessional at a Greek Orthodox church and says "Forgive me, Father, for I have sinned...", and it turns out the Priest is actually Q, who says "That's putting it mildly, 007.".
This is the central plot of Alfred Hitchcock's film I Confess.

Literature

A minor plot point in Memory Sorrow and Thorn is Shell-Shocked Veteran Camaris' confession of his ~~sins~~ role in the plot to Father Strangyeard. Camaris is asked to do this because he refuses to reveal his secrets in the open, but the heroes desperately need to know if what he knows has any bearing on their struggle with the Storm King. Although Strangyeard confirms that Camaris knows nothing useful, the confession devastates him, making him admit that for once, he understands why people might wish to drown their sorrows. It's also a convenient narrative way to hide Camaris' secret until after the climax, when Josua (to whom Camaris also confessed) reveals it.
In The Gadfly, the protagonist loses all faith after he learns that his confessor leaked the secret to the police.
In Earth (The Book) this is double subverted. There is a transcript of a confession where the person confesses to: speeding, a hit-and-run, hitting a cop in the hit-and-run, robbing a bank, and murdering people in the robbery. This causes the priest lots of discomfort. It seems completely played straight, until you realize it was transcribed. This is lampshaded, saying the transcriptions were for 'insurance purposes.'
In E.F. Benson's "The Hanging of Alfred Wadham", a murderer confesses to a priest—to be sure that the priest's the only other one who knows an innocent man is going to die for the confessor's crime—and that he can't do anything about it.

Live-Action TV

A Catholic Priest hears a confession of a crime in A Touch of Frost that causes him some real difficulties.
Subverted in Leverage, when Nate (a mostly-trained former priest) uses the sanctity of the confessional to achieve his aims as a conman.
- In his defense, the plan he came up with as a result hinged on providing the confessor a chance to do the right thing and make the confession in public.
CSI - A Catholic priest, bound by his confidentiality, at least tries to steer the investigators in the right direction.
Poltergeist: The Legacy: A priest denounced a serial killer that confessed not only his recent murders, but also some future ones. He later rationalized it because the killer didn't actually repent for his crimes.
Highlander the Series - On at least two occasions evil Immortals use confessions as opportunities to gloat and the priests involved are unable to report their "confessions."
Ian Hislop has claimed in interviews that at one point during his long-standing feud with Piers Morgan, his vicar told him that the Daily Mirror had called wanting to know if he'd confessed "anything good."
In one episode of Law and Order, a murder case hinges on whether a priest will break confidence and finger the man who confessed to him. And in case the decision wasn't hard enough, the victim in the case was also a priest.
Father Mulcahey has had to figure out ways to resolve issues that he learns about in a confessional without violating the seal of the confessional on multiple occasions on Mash.
Zig Zagged in the Italian series Don Matteo. The title character, a priest and amateur sleuth, is convinced that a man is a killer, and the man gets so annoyed of Don Matteo investigating that he goes to him in confession and says, basically, "Yes, I killed that bastard, and now you won't be able to do anything about it because of the secret of confession!". Don Matteo is stumped for a bit, but then tells the culprit that he would gladly be excommunicated if it meant putting the culprit behind bars. Then the culprit sees Don Matteo talking to the police and attacks him with a hammer, screaming "I'll kill you so you won't tell them I did it!". Don Matteo, however, wasn't telling on him, but simply having a nice chat with his policeman friend...
An episode of Murder, She Wrote opened with a woman confessing to murder, and the priest having to decide what to do about it. Of course, this being Murder, She Wrote, the woman hadn't committed the murder after all.
In the Diagnosis: Murder episode "Confession" the killer confesses to a priest and then frames the same priest for the crime.

Western Animation

In The Simpsons "Who Shot Mr. Burns? Part 2" Smithers goes to confession.

Smithers: Father, I'm not a Catholic, but...well, I tried to march in the St. Patrick's Day parade. But anyway, I've got a...rather large sin to confess. sniffles I'm the one who...shot Mr. Burns!
Wiggum:( pokes head out, cocks gun) That's all I needed to hear! Boy, this thing works great.

Real Life

This has been invoked by many Russian tyrants, most infamously Ivan the Terrible, but he's not the biggest offender for this in Russian history. In late XIX and early XX centuries, before the Red October, Russian imperial police required the priests to report any crimes, criminal or political, they learned about in confessionals.
There was a scandal a few years ago where a priest was appointed to a Polish bishopric and was discovered to have revealed secrets he learned in the confessional to the secret police back when the Communists were still in power.

Physician Examples

Anime and Manga

Black Jack has run into this a few times. Of course, since he's an unlicensed surgeon, he usually only worries about patient confidentiality when it suits him, but he occasionally finds inventive ways around it... like charging a bank robber all the money he stole for a life-saving operation, and then turning the money in to the police.

Comic Books

In Judge Dredd there was a serial killer that had a psychiatrist who was trying to cure him that kept a confidence in this way. If he felt guilty about it however, he didn't say so.

Film

In the Billy Crystal/Robert deNiro film Analyze This and presumably the sequel, a mob boss-type character gets his psychiatrist involved in his shady dealings this way.
In Grosse Pointe Blank, the main character (a hit man)'s therapist tries to explain the loopholes in confidentiality and being required to report it when/if he knows his patient is going to hurt someone. The patient assures him that it's fine, he understands, and he doesn't want to make things difficult for him, and anyways he knows where the doctor lives…

Live-Action TV

Referenced at the beginning of The Sopranos: Tony Soprano's psychiatrist tells him that if he confesses to her any serious crimes, or suggests that someone is in physical danger (eg, that he intends to kill someone), then she is a mandated reporter and has to pass the info on.
- Also they used Doctor-patient privilege to have meetings with Junior in his Doctor's office while he was on trial, since the government could not wiretap the Doctor's office.
In an episode of Dream On, Martin is dating this wonderful new girl that he hasn't slept with yet (he's trying something new), but it turns out she's a client of Judith, Martin's therapist ex-wife, and she becomes homicidal after sleeping with someone. Judith eventually puts it together, and shows up at Martin's apartment just after they've consummated their relationship and as she's about to kill Martin.
Non-criminal example in Scrubs. JD is smitten with a girl and unintentionally agrees to treat her boyfriend. He diagnoses a man with an STD and he confesses that he probably got it from a girl he was seeing on the side, then invokes doctor-patient privilege to force JD not to share the diagnosis or the fact that he is cheating. JD has to choose between warning the girlfriend or his professional ethics.
- This is averted in some jurisdictions, as sexually-transmitted diseases need to be reported to the local health authority and/or the patient's sexual partners. In this specific example, the girl and her boyfriend had not had sex yet so JD couldn't use this loophole to tell her anyway. Fortunately, she figures it out before sleeping with the jerk, when her coworker (who her boyfriend slept with) develops the same symptoms and gets diagnosed with Gonorrhea too.
In an episode of Grey's Anatomy, a woman confesses to purposefully ramming her car into her husband. The two doctors listening point out they only share confidentiality based on medical information, not criminal activities, and she's arrested.
In an episode of Frasier, the eponymous doctor finds a loophole in the psychiatrist/patient confidentiality agreement by becoming a patient to his brother (also a psychiatrist), allowing him to tell his brother his patient's troubles.

Video Games

In Heavy Rain, Ethan Mars' shrink initially refuses to talk to the police, citing doctor-patient privilege. Detective Blake, a definite Bad Cop, just beats it out of him, and the stuff from Ethan's psychic evaluations turns out to be pretty damning.

Attorney Examples

Film

Part of the ending of Primal Fear: A murderer who escaped justice boasts to his lawyer that he committed the crime but the lawyer can't tell anyone else because of attorney-client privilege.
And Justice for All involves a lawyer being politically blackmailed into defending a judge (who he despises) who's been accused of beating and raping a young woman. At one point the judge confesses to the crime.

Live-Action TV

One episode of Law & Order has a lawyer who refuses to reveal the location of his client's victims' bodies. At first it seems like he wants to avoid introducing evidence that would damn his Complete Monster client. Then the client is convicted and they assume he doesn't want to get in trouble for break privilege or is trying to impress his bosses. So they basically have a judge tell him point blank he won't be arrested unless he doesn't tell them. It turns out that he really doesn't want to break attorney-client privilege on principle, and he goes to jail for contempt of court.
Subverted on The West Wing, where it turns out the the White House Counsel is not actually the President's attorney:

Bartlet: Well, Oliver, it really boils down to this... I'm going to tell you a story, and then I need you to tell me whether or not I've engaged 16 people in a massive criminal conspiracy to defraud the public in order to win a presidential election.
Oliver: Okay. [...] Okay sir, uh... before we go any further, there's something that I want to make sure is absolutely clear.
Bartlet: What's that?
Oliver: You and I don't enjoy attorney/client privilege.

White Collar: Neal exploits the hell out of this when he gets framed and arrested in season 1. Since his attorney is his partner in crime Mozzie, they can use attorney-client privilege to keep the FBI from monitoring them while they plot Neal's escape.

Visual Novel

Ace Attorney ran into this, where one of the protagonist's clients actually is guilty and admits it, but the hero can't tell anyone. The assassin who was hired by Engarde kidnapped Maya and threatened to kill her if Phoenix didn't get Engarde off the hook, so he couldn't say anything even if he was allowed to, or even drop the case.