The difference is that every prisoner in a private prison is an income source, so every law that creates fewer prisoners (such as drug legalization) is a loss of income. Are all public prisons run the same way? That is, are the prisons funded on a per-prisoner-per-day basis? Or do they get an overall annual budget, and then have to spend it as best they can to house the prisoners they're sent?

In the former case, harsh laws are good for them, same as the private scenario. But in the latter case -- fewer prisoners on the same budget -- more permissive laws would be an unconditional win. I don't know which way it actually works, just spelling out an unstated assumption that's being made upthread.