Most people are not in private prisons (< 10% [0]), even if there shouldn't be any at all. Of course, there are still many "contractors" and "vendors" (phone service providers, food vendors, etc.) in public prisons which grift everyone.
[0] https://www.sentencingproject.org/reports/private-prisons-in...
Public is when the government holds the wallet and pays vendors. Private is when they give a wallet to someone else to pay the vendors.
After becoming familiar with the reality of the cost inflation of (in my case local government real estate) development projects vs private I chalked it up to graft, incentives, and mismanagement.
Actually your comment is probably more correct - adds a whole step to move the wallet. Misaligned incentives and mismanagement are probably more equal across public/private than we like to believe
I'm being a little bit facetious. When the government actually owns/operates the labor or equipment they can do a lot more. In the prison example state COs are certainly better than rent-a-cops.
It's just unfortunate that's how most administrators work. The traditional debate about public vs private usually focuses on different tradeoffs and incentives of the public - but if they are just paying market vendors it's greatly diminished.
Who signs the paycheck of the warden and the officers is another way to differentiate.
I agree the officer story is the most significant difference, with state COs being more like police - likely to be well trained, have a long term stake in the career, and having somewhat of a social service culture.