Obviously, it's the state or some other public authority that foots the bill. Private enterpreneurs sure as Hell aren't going to run prisons for free, that's for certain. At which point there is a very good reason to ask, why not just cut out the middleman and have state prisons in the first place ?

Then again, shameless and often irrational profiteering is the the fundamental bedrock of Amercian values anyway far as I've seen.