My take: a society that intends to function witout a prison system and police force, must obviously place its main bet on prevention. Every time a person doesn’t attack another person, a cop isn’t needed.

However, even if incident rates can be brought low, violence will probably not disappear - so even an anarchist society would need procedures for dealing with crime. Some guesses:

• Instead of a permanent class of cops, people might rotate through guard duty; some percent of people might have the education required to investigate crime, but to prevent the potential for a repressive system, they might get called up randomly for brief periods of time (nobody should be a fulltime cop or judge).

• The emergency response system could be supplemented by a decentralized system of the nearest qualified person responding. Responding to a fire without a fire truck may be not very effective, so obviously one would keep fire depots. Responding to illness without an ambulance could be ineffective, so emergency rooms would probably still exist. However, for a first response to individual violence, not much is needed - a random person with functioning self-control, ability to gather information and reason, a means of communication, a can of pepper, a flak vest and a gun is actually more than typically required. Ensuring impartial response would be the tricky part - a local person might be the quickest to arrive but not impartial enough to investigate. For stopping violence, I believe that locals would be a great choice, though. Not inclined to gun anyone down.

• Investigating what happened might go a bit differently, with more than one institution gathering evidence. There might be less distinction between a civil complaint and a criminal case. Alternative courts might overlap in juristiction and recognize each other by establishing alliances. There might be no hardcoded law. Establishing what truly happened beyond reasonable doubt would probably still happen, though.

• Finding a person who tries to evade justice might work differently. Instead of few people having great powers of surveillance, in anarchy a lot of people would have limited powers of surveillance. It might resemble InterPol procedures (“you receive a warrant -> you check that it wasn’t sent by crazy authoritarians -> you check if the crime is considered a crime locally -> if OK, you proceed with searching for the person”). In anarchy, a person could evade justice considerably longer and more persistently, at the cost of moving, losing their network of trust and becoming an outcast.

• Resolution of the crime would likely differ. A guess: local community would help and compensate harm to the victim (or in case of deadly crime, look after people close to the victim). Meanwhile the criminal might be required to compensate to the wider public, undergoing training or treatment, making themselves useful to a greater degree than usual and thus demonstrating that they can reform themselves. If the victim doesn’t agree or the criminal doesn’t follow through - there might be a default judgement in store: the criminal might be expelled, communities in the region might be notified of the expelled person and their crime, and the judging community might declare that it will defend and harbour the first person who takes proportional retribution.