There are some seriously deranged people out there. You cannot just release them into society. I think you're looking for solutions to problems caused elsewhere.
The most deranged people should be in well regulated mental hospitals, not for profit prisons. It's insane to acknowledge that there are people too maladjusted to operate in society, then send them to the one place where you know they'll get no help.
The vast majority of prisons are run by governments, and even private prisons are regulated. Using hospitals as a substitute for prisons will result in harm to treatable individuals, as well as more stigmatization of mental illness.
As well as treating mental hospitals as potentially permanent prisons even if the crime itself has a finite sentence. In a world with no bad actors it might make sense but it only takes one bad psychiatrist (or psychiatric facility) to punish a perfectly sane individual indefinitely (or drive them to actual insanity after prolonged confinement and unnecessary psychiatric treatment)
