I would argue, too, that corporations want to maximize profit. Killing people costs money and generates bad PR. Working camps might actually be worth their invested money. Of course horrific working conditions will eventually leak out and generate bad PR, too. So AA(A) corporations might individualize punishments and use every criminal towards their best capabilities. White collar criminals might get brain-washed (psychotropic treatment) and work in their field of expertise. Especially ruthless corporations might also hook people up into the matrix, while their bodies are paralyzed and force-fed. So your mind can work 24/7 (well, almost) for the reminder of your life
But if your mind cannot be put to work, I would assume that at least smaller corporations that have no use for you, will sell or lease you to KE etc. and they will put you to good use. They have the facilities for that, possibly hooking you up, throwing you into the mines somewhere or use you anyway they are pleased. The PR-problem remains in place, though. Too much inhumanity only makes matters worse; detainees start revolting, tight security costs money, prisoners sabotage their work, investigative reporters sneak out trideos etc.
The corporations cannot afford to handle prisoners like the Nazis did their forced labor workers from the USSR etc. People need a perspective, otherwise they stop collaborating. Sure, for a Ukrainian in Nazi-occupied Europe forced labor in a factory was a huge opportunity to stay alive. But only, because the Nazis didn't have to care for their image in the rest of the world. The corporations have to, so if imprisonment is better than life on the street, what will work get you? Work needs to be an incentive. Maybe corporations offer limited SINs after 25 years of hard labor? Why not? You still need to make it, play by the rules, acquire education, play even more by the rules etc. Incentives also generate competition between prisoners, thus breaking them up into fractions. That's good for population managment and helps avoiding sabotage, unrest etc. The Nazis used camp inmates for low level administrative duties. They might think they were on the fast track to survival, but they couldn't be sure.
In the end, corporate prisons have the same problems as public prisons, just more so. There is almost no motivation to cross-fund them. They need to be productive, because they supplement resources. If you're R&D department is in the reds, you can crossfund it and hope the guys eventually come up with something worth the money. If your strip mine costs more than you get from selling the coal, it's hard to justify keeping it open. Unless liquidiating the mine would cost you assets or break contracts somewhere else. But then you should start renegotiating contracts and re-organise your corporate structure.
