All major coorporations have a security department that investigates losses, interviews witnesses, gathers evidence, and hunts for the perpetrator. It is only a small step for PI's to move to the other elements you describe.
Government holds no special monopoly on it.
Private citizens are, under our constitution, already able to arrest for felonies. Not as the game is played today, thanks to the securing of monopoly powers of the State. And, under a truly free society, there would be a lot less rape, murder and robbery, as people would truly be able to defend themselves. Also, skip tracers retrieve people at gunpoint and return them to the venue that they're supposed to be in, based on a warrant. If we were to allow PI's to go to the judge, show the evidence and get a writ from the judge that the suspect is wanted for questioning, then a similar process would work rahter nicely.