FYI the libertarian solution for private prisons isn't just "let prisons be operated by private enterprise", because to them that's like say "Liberal position on death penalty is that all the executioners must be liberal".
Libertarian solution for private prison would involve a full overhaul of justice system. Our present justice system is based on roman law. The idea being, that law is what is being enforced by the King or the Senate/Republic. This means that the law must be codified otherwise the enforcers could enforce anything and call it a law. This needs the law to be more centralized.
Under this tradition, law stops being something objective, and starts being subjective (that is 'anything which can be enforced is law'). Take for instance when Nazis made it illegal for Jews to own businesses, most of us would know that there is something unjust about it, even if the Nazis have full capability to enforce it.
Libertarian law is more similar to ancient Irish Brehon law system where the law is more decentralized. There is no central authority which grants enforcement powers, rather the individual interaction among individuals is what they must bring enforceability.
How does that play into the prison system? Under the statist law, bad actions must be prevented, and it's done at the threat of retribution and punishment. Bad people must be punished for their bad actions.
Under libertarian law, bad actions must also be prevented, but the aim is to deliver restitution, to make the victim whole.
The person who commits a theft or murder, and is proven to have committed the crime, then he must pay back the victim or victim's family. If he can't pay it back, then he must submit himself to an institution where he can work and pay the victim back. Private prisons are these kind of places.
Now of course there are numerous other questions which arise, like why would anyone ever submit themselves to such an outcome? Or what if you are rich, could you then keep committing such crimes?
I would highly recommend reading this short book to answer a lot of those questions: https://mises.org/library/chaos-theory
If you don't care to do that, and/or are more interested in reading about ancient Irish law (who had the same problems as a libertarian system would have):
The Brehon Laws: A Legal Handbook
Lost Laws of Ireland by Catherine Duggan
Or if you like fiction, then you may try this: [Fiction] My Lady Judge by Cora Harrison