There won't be any. There will probably be a very small number of houses of confinement, as below, but prisons as they appear today will not exist.
That's because in the coming Zero Government Society there will be no government, and so no laws, and so no criminals, and so no punishment of criminals; and prisons are places of punishment.
There will be a vibrant, free-market, competitive and profitable justice industry, mostly concerned with resolving disagreements among parties whose contracts weren't worded well, but it will deal also with the few who, despite having learned that every person has the right to own his own life and property, nonetheless violate someone's self-ownership and thereby commits what I've called a krime. Damage, caused deliberately; for which restitution is required.
The normal outcome or verdict from a free-market court case will be an order, under which B compensates A for damage he caused, but without their having agreed the amount of compensation due. That process is almost identical to the one used in medieval Iceland, during the two centuries following AD 850 when Norwegian refugees founded that society on the unoccupied island. A court ("ting") met once or twice a year in the location pictured, to settle such residual disputes.
The strong reason why B will comply is that if he doesn't, the fact will be made public and he will lose a serious part of his good reputation; and if that happens, few will trade with him. In egregious cases, few will even sell him the necessities of life; he will starve to death.
There will also be a few cases in which the party at fault ("B") violates the rights of several people on purpose, repeating the acts of aggression; he's a serial killer or rapist or thief. I think those cases will be rare indeed, though cannot say they will never occur. One reason they'll be rare is that whenever a kriminal aggresses in such ways, he will run a heavy risk of being shot dead - by his usually well armed victims, or by well-armed passers-by. To commit such acts serially means that heavy risk will be run repeatedly; the probability of surviving multiple aggressions will be tiny.
But when such rare survivors come before a court, it may order not only restitution, but confinement; not as punishment, but to protect future probable victims.
The confinement house will of course be run for profit, and will impose no hardship on residents other than the prevention of escape. They will be free to engage in trade with the world outside, and will be charged for accommodation and food, and of course discharge restitution obligations ordered by the court. Persons of low ability will be provided with simple work, so that they'll have at least enough to pay those charges. If they refuse to perform it, they will starve to death unless someone donates help. But others with rare skills (eg software writers, novelists) will be free to earn large sums and spend them as they wish (except on travel!) - for example, they might order the delivery of gourmet food, or rent luxury rooms, or donate help to fellow-residents in need.
Conjugal visits? - of course! Wives, girlfriends, whatever. Come to that, some might have partners living with the resident every day, coming and going freely (with appropriate ID.) The only purpose of these houses will be to prevent a serial kriminal harming the public. His harmless friends and relations will be no business of the justice industry.
How long will such people be confined? - no longer than they remain a danger to others, of course; the confinement is not punitive. Can it be remedial, however? I hope and think so. Why kriminals go on repeating aggressive behavior, destructive to others and themselves, is a mystery which in time may well be solved. If the brain needs a re-wire and surgery can do it, the treatment will be provided, for a fee, and a court will be petitioned for a release of the confinement order. Who will pay? - the perp seeking the release, and that does mean the richer ones will be treated and released first. That's always the case, with any new product or service. As soon as the treatment is proven effective, however, courts can be expected to prescribe it for all serial kriminals, instead of confinement.
To today's "Lawn-Order" classes, the above will at first seem outrageous - far too soft on criminals. We have to remind them that there will be no criminals in the ZGS, so it will be neither hard nor soft. This mistaken impression can also be found in libertarian circles, always because people haven't thought enough about how the ZGS will work. Even Murray Rothbard never quite emerged from the common belief in punishment, and that's one of the very few respects in which he was mistaken. His magnificent contributions to economics, history and philosophy remain unrivaled, but I think he never gave much attention to how the justice system will work in a free society.
Neither is this ZGBlog a comprehensive account of how a such a system as a whole will probably evolve after government has vanished; it focuses just on the one aspect, prisons and their absence. To see how its other parts will likely blend together - detection, apprehension, disposition of krime and the important part played by insurance companies, see Chapter 7 of my Vision of Liberty.