Criminal cases aren't a substitute for civil suits, not for copyright... or for any other type of loss.
People generally do have to pay their own way to bring a civil case to recover for damages in a copyright infringement case... or any kind of case.
The fines/jail time typically ascribed by a criminal case do not go into a victims bank account. A criminal case is between the government prosecutor and the defendant. The copyright holder wouldn't even be a party to the case.
Many states do collect restitution funds from revenues generated by the work of encarcerated people, and those funds do go to victims. I don't know that that applies to copyright infringement, but it is possible to get some recovery from criminal proceedings.
If a criminal case ever happens, it is a possibility that restitution can be awarded. But generally, if somebody's infringing your copyright and you want to seek damages, you need to bring a civil case yourself. Well over 99% of copyright cases are civil.