I'm thinking--just off the cuff--that public safety (against a bad cop) would out-weigh the secrecy of personnel files.
Plus, adroit use of the sunshine laws might pry out some information. For example, VA's sunshine law (freedom of information act) expressly exempts personnel information. And, yet, it requires the custodian of the information, when refusing to provide a copy of a record within his discretion allowed by law, he is required in his refusal letter to you to state the size of the withheld record.
So, instead of asking for "the written application of Officer Jones", you ask for "those pages or part of pages that show Officer Jones was fired from any other PD."
This takes subtlety. You've got to word this carefully. You can't say, "pages that show he was fired or allow to resign". You just get a refusal that explains there are four pages of application responsive to the request for termination or being allowed to resign--and everybody is allowed to resign to move on to a better job. And, you can't tell which is which. In those four pages was he fired, or allowed to resign? Which times?
Maybe you have to ask, "those pages of Officer Jones employment application which detail being fired by an earlier police department." And, then after getting a refusal on that point--let's say the custodian of the records does not say, "There are no such records" and instead says there is one page, withheld under the custodian's statutory discretion. Ah-HA!!! Now, you know he was fired by an earlier PD. Now, you have to send another FOI (freedom of information) request narrowing the scope. Maybe you have to send a total of four requests as you narrow the scope, step by step. For example, "...that page or pages of Officer Jones application or employment history verification investigation that show he was fired for violating department policy regarding the rights of suspects". My point is that you have to be slick, and painfully precise.
Side Note: If personal experience is any example, expect the police to obfuscate, mis-direct, pretend to not know the requirements of the sunshine law even when you expressly cite the exact paragraph of the statute, or outright refuse in violation of the law. Literally--no exaggeration--of all the FOI requests I've sent to police, once they figure out what I'm up to, they start dodging 75-80% of the time. I've had a police lieutenant in one of the most self-proclaimed professional police departments in the country come all over stupid like he can't read and understand a very, very precise FOI request, and respond totally off the point with information not requested when it involved a cop. In one sense, treat your FOI request as a genuine FOI request that will get you answers. But, in the back of your mind, be ready to treat it like a dance, where all you're doing is really backing the PD into a corner and gathering evidence for a lawsuit. That is to say, almost expect the PD to be dodgy, to the point where you don't really expect them to comply with the sunshine statute. Assume that all you're really doing is building a paper-trail of bad faith non-compliance with the sunshine statute in preparation for a lawsuit. In one sense, be ready to take it as though you don't really care what their reply is; you know you're just building a paper trail proving bad faith dodging for a lawsuit. But, hey, that's just me. I've only been through this a dozen times with police departments. What do I know?