My experience in the prisons is limited - I suppose - I've worked in three and volunteered in one. However, in all of the prisons that I've been in there was no difference in treatment/care of inmates between public and private prisons. The same abuses happened in both.
The only difference was that the guards were paid better in state prisons. All prisons provide heath care - btw - though it's true, it's not always very responsive. And, it's also true, they don't have access to the same kind of care they would on the outside (so in many cases, it is dangerously deficient).
The problem, ime, isn't public/private - it's that in both there isn't a mandate to work with the men/women on treatment. Instead, the mandate is security. In the past, when we provided better educational options for folks - there was considerable backlash. In the prisons that I worked in, men were arbitrarily moved from prison to prison and program to program (so if they started in plumbing and HVAC, for instance, there was a very good chance they'd never finish). And that was the VERY few that qualified for that program. My point is that even the educational programs that they have - suck - because no one cares if they really get the certificate. There were plenty of GED classes - though (some teachers, often classes taught by other inmates, which could be okay - but.....it wasn't. It was aimless and warehouse-y).
The men cared. But the institutional mandate was 'safety and security'. And that's our fault (the public). We don't want them getting degrees and learning skills that 'we can't get for free' outside of prison.
Also, the federal prisons rock. They have tons of money and lots of programs. Really, they are models and it's because they have programs and opportunities. Has nothing to do with that public/private thing except the FEDS put money into the institutions. Money matters.