As i have come to understand what may happen regarding crime and punishment in a an-cap society is,if a thief gets away with some of your goods/property,and you can prove its the person in question, you can use follow up force to get it back,you have not initiated the forced, you can sue this person.But a problem might arise when you confront the this person with or without your local private security force,and try to obtain your goods,because his security force or other witnesses don’t know who the original aggressor is,unless the person in question has a bad reputation.You can still sue this person but if you have no real proof they could walk,and that is just the way it is.The person in question might not turn up to court if he so wishes but then hopefully this would be bad economically in some way to this person.But if found guilty they have to pay restitution,compensate you for the stolen goods ,compensate you for your trouble and time,and pick up all court costs.And this does not contravene the NAP.
As i said in my last post,if there is proof of any crime you can sue the person concerned.But you cannot force them to go to court,if they did not turn up to court it would not favour their defence,and hopefully this would have some sort of effect on their lives economically and in society in general.
I would like to touch on proportional punishment.I believe in a libertarian society the punishment should fit the crime,IE capital punishment only in the case of murder.And that is still only the concern of the family of the victim.The murderer could still buy his way out, if the family of the murdered person agrees a fee.
"Shooting the criminal in the face" even if that person is steeling some bread and milk from your store unarmed??Shoot them in the face? A young boy steeling a bar of chocolate or some chewing gum,shoot him in the face?I don’t think so,that would make you the aggressor.