The trouble is that private prisons have led to high numbers of illegal convictions. I don't have a cite, but a local (very conservative) juvenile court judge shared a bunch of material about a major scandal in Pennsylvania where a whole bunch of kids were improperly "adjudicated"
When the rights of a large group of people are limited by statute, they become substantially more vulnerable to abuse by the system. When the system has a strong profit motive for imprisoning more people, well, you tend to get people unjustly imprisoned.
I'm not against for-profit prisons or lobbying by them, but it is important to recognize the potential problems and to have excellent oversight and review systems in place. I don't know enough about the law or AZ state gov't to know if that is in place.