Now the three groups aren't equal, but they do exist on the same scale. For profit prisons are, IMO, the worst while the other two groups are of debatable ranking depending upon the size of the relevant organizations.
Maybe there's an interesting thing to say here - some people people are against for-profit prisons because they think they encourage imprisoning more people as it creates more profit.
But if you have a non-profit, government-run prison in your local area, you may still be encouraged to imprison more people as it creates more profit for the local suppliers to the prison.
So yeah it doesn't matter if the final user is for-profit if you think there is a risk that profit is driving something you don't want.
This is a gross simplification that erases the very real differences between the incentives of a government vs. a for-profit company. Not only is the government accountable to its electorate, but a government that wants to save money can take a much broader approach - for example by funding access to mental health services - to keep people out of prison. Prisoners are expensive to keep locked up!
In the final analysis, all prisons are non-profit. None of them generate any value. They're all funded with taxpayer money that could be spent elsewhere. It's just that in the case of private prisons that money goes to a company that has no interest in saving the government money by reducing the prison population. Sure they'll try to run the prison efficiently, so they can pocket the difference, but they want as many prisoners as possible to get a bigger check from Uncle Sam.