The extent of my expertise in this area is being a big fan of NYPD Blue and watching them cover similar things over the years <g> but based on everything I've seen, the police would first conduct an investigation.
If the apprehender were a police officer, it might get bumped up to Internal Affairs, but if the apprehender was a civilian (as in your story), it would likely be handled by police detectives as a regular assault charge. The police would look into it, gathering whatever evidence they could -- taking statements from all the parties involved, comparing the details given by the suspect with that of the apprehender, interviewing witnesses, looking into the background of both parties (to see if either had a history of assult or bringing false charges), etc. They would also want to make sure the suspect's injuries are consistent with his story.
Once they gather the evidence, the police would make a judgement as to whether the accusation had any merit and would discuss those findings with the District Attorney to see if formal charges should be brought. The DA (probably an Assistant DA, actually, unless it was a really high profile case -- think Mayson Drake vs that sleezy guy in TPvLL who wanted to run for higher office) would evaluate the evidence to determine if the police actually had a case. Her options would be to agree that they should arrest the person(s), tell the police that they needed more evidence to get a conviction, or agree that there is no merit to the case.
BTW, this stuff only applies to criminal charges -- as we saw with the OJ Simpson trial, the rules are different for civil cases and even if the police declined to bring criminal charges against the apprehenders, the suspect and his lawyer might decide to file a civil complaint, asking for money for pain and suffering and punitive damages.
Hope this helps!
Kathy
