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I doubt they admit to anything, but having some information that helps lead to the recovery of a valuable item could be worth something. To be honest, this is the first I have heard of the practice, but it sounds plausible to me.
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There is a plotline involving this in the first season of The Line of Duty.

It opens with a guy who's complaining to the police about the frequent robbery of his house. They view him as a nuisance and try to respond to his complaints as little as possible.

Eventually someone is apprehended committing a different crime in a different location. An officer visits him in the hospital (there was a mighty struggle before he was caught) and asks if he'd be willing to lower his sentence by potentially resolving some other outstanding crimes that the police are investigating.

She reads off a list of crimes, and he responds "yeah, that was me" to everything she says. One of the items is the several robberies of the guy we met earlier.

It's win-win; his sentence is lowered for doing the police a favor, and the police report a higher solve rate.

After that, the original guy who was always brushed off suffers another robbery. Since he happens to be home at the time of the latest one, he is severely beaten.

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Plea bargaining. Green river serial killer admitted to more killings (and cooperated to recover some of his victim’s bodies) in exchange for avoiding the death penalty : https://en.wikipedia.org/wiki/Gary_Ridgway#Plea_bargain,_con...
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Crime gravity. If able to provide the solution to a crime that is considered more serious(or of greater value to the prosecuting entity's purview).
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So the lesson here is if you are a criminal - commit multiple crimes, hope in case you get caught then it is for a lesser crime, so you can trade secrets of a bigger crime for leniency?

I wonder what victims have gotta say about that...

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> I wonder what victims have gotta say about that...

You only let them say anything every four years or so. And then you set aside their complaints by saying some of them are made up.

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"Hey, I don't have any idea how it got there, but I heard some guy talking about a famous stolen painting that's stashed right here in this neighborhood. Apparently a Renoir."
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