I think the issue with regard to privatized prisons (and perhaps would-be privatized police) is that there's a profit motive. A need to keep profits up. Whereas public prisons can take the hit --ie. the losses are incurred by state coffers. A private corp can't take that economic hit as well.
Private police, though ... in an era where early death is relatively uncommon vs. omnipresent, what's their incentive to take risks beyond what they're paid for? That what can be problematic, there's many reasons self-help is outlawed, but in the examples you give there's no immediately obvious problem.