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.
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. While these injuries are undoubtedly tragic, they are far down in statistical noise, and if similar figures were quoted about guns, say, that out of whatever many million guns there are in the US, fewer than a hundred will be used to shoot a child, he would rightfully be the first to hoot in derision. 22 to 36 chances of injury out of 3.6 million are odds that almost anybody would find highly favorable to non-injury. I have, in general, very little sympathy for faith-healers of any stripe, and quite enjoyed an anecdote in The American Atheist magazine some years ago, where a woman who'd grown up in the Christian Science church told about how, at church one day, one of the elderly ladies who made up most of the congregation said that she had witnessed a motorcycle accident, and, trotting up to the victim as he lay there groaning and bleeding, told him that there was no pain and that his injuries were all in his mind. One kid, who had already begun to walk and talk, reverted and didn't walk and talk again for more than a year.