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Can’t speak as to how they do it now, but bounty hunting wasn’t really… formally a thing in the Roman period, because of the weird way Roman law worked. Basically, most law was civil rather than criminal, and winning a case in a civil trial basically meant “You have to cooperate with your accuser for the fine levied, or he can declare you bankrupt and auction off your entire estate.” Thus, if you really wanted to track someone slippery down and force them to trial, you could pay someone to sniff them out and serve them with summons (or, depending on how far along the trial was, manhandle them into prison to force attendance of the trial), but that would be purely a question of “How much is it worth to YOU, the victim or plaintiff?”

Criminal law was basically restricted to murder, corruption, conspiracy, and crimes against the state; and if something was bad enough to cause the Roman government to get involved, chances were pretty good the local enforcers - or worse, the soldiery - would be sicced on you. I doubt they recieved any bounty for the job, except ad hoc. A person could be outlawed, but that just made it legal for them to be killed, it didn’t offer any rewards in-and-of itself.

Depending on the crime and the period, those assisting capture and conviction might be rewarded with a portion of the convicted’s estate (with the rest seized by the government).