Actually we work with (and are notified by) PayPal's fraud dept (and other providers as well) whenever there is a questionable transaction.
As for cash and the like, usually local LEOs deal with it and, unless it was us reporting it, PC likely will not be contacted.
To answer the OP's question.. what would we do..
well If we found out it was stolen/fraud/etc, we'd try to get it back to who it belongs to and, well, the user that used it would likely be banned for fraud and their details could be turned over to the fraud peoples.