agree it's going to depend on the relationship between the agencies and how the 2nd one knew there had been an application at the first (they might share an ATS for example). And then it might depend on that authorization form.
Unfortunately the OP's spouse chose to write different answers on the two PHSs....it might also depend on what she disclosed vs left off. BG Checks for law enforcement/government positions are meant to weed out applicants with issues....
One of the minimum requirements of a job in law enforcement is that applicants be of good moral character and that they possess the traits of integrity, honesty, dependability, thoroughness, accuracy, good judgment, initiative, and resourcefulness (to name a few) as determined by a thorough background investigation. To this end, applicants sign a document authorizing the release of records relative to such an investigation, which includes access to prior background investigations. in doing so, they release persons cooperating with the investigation from any liability for compliance.
Reviewing prior background investigations is a measure of integrity, as some applicants will lie and change answers they think might disqualify them. It also allow assess to both positive and negative information about an applicant another police department may have discovered.
To apply for a job where honesty is paramount and then complain because one was disqualified for dishonesty is amusing at best, but disingenuous in the end.
There is no out or limitation on these forms nor can they be modified to say you can have this information but not that. To do so effectively limits the ability to perform a thorough background investigation.