Well it's not like all the non-privatised systems (think of Rikers) have covered themselves in glory of late.
Are there studies showing better outcomes in state-run systems?
Clearly we incarcerate far too many people and for too long, but that's a somewhat different question than the private/public question. Of course I hold no brief for the private prison companies lobbying against shorter sentences - that is disgusting. The whole prison system, public and private, seems rotten.