This is a pretty hot issue in the last few years, and there's various considerations that should be taken into account. When a prison is privatized, the company which manages it provides a service. This service can be better or worse than services provided by the state, and that would depend a lot on what kind of prison it is, what industries it relies on, etc. While I'm not completely opposed to privatization, what concerns me is that this might lead to merely economic considerations which would not do much to reduce crime. On the other hand, with proper regulation, it may free resources to fight poverty, encourage social welfare, etc. - not that it's guaranteed to work that way for all institutions.

The institutions I've come across were all public - civilian and military prisons in Israel, and San Quentin prison in California, so I haven't seen a private prison firsthand, only read about them.

In Israel, the decision is made by the correctional services (I believe the judge can make recommendations). In the US, since prisons are classified according to their security level, sending someone to a certain level of prison can be plea-bargained.