The justice system still sets the parameters and manages the procedures by which sentences are handed down. If a court determines you're going to prison for 10 years, you're going -- the private company manages your environment for that time.
And given that private parties also manage your medical care, hospital stays, air travel, car manufacture, work environments, home building, and thousands of other mechanisms and environments that determine your safety and quality of living, I struggle with finding a moral difference when it comes to prisons.
You could argue that these other things are voluntary and prison isn't, but it's involuntary even if run by government; most inmates would opt out either way.
And I think it's highly naive, this distinction between private companies as greedy, interested parties and government agencies and functionaries as objective arbiters of truth and goodwill. Either way you have ridiculous salaries, internal power struggles, marketing/perception, budget committees, and cynical people inured to the hard realities of the lives they affect.