Thank you for the opportunity to help you with your question!
They are both under oversee of local ,state or federal governent with only private one beibg handled by third party contracted by the state or federal government. Most people argue that private jails are cost effective and space efficience.
They typically house same prisoners as most of them are transfered from public jails. However ,top criminal targets like terrorists are mostly housed in public jails.
It is a good idea to privatise jails. It tries to easy congestion in public jails and also unburdens public jails from ecxess expenditure since private jails are space and money efficient.
Please let me know if you need any clarification. I'm always happy to answer your questions.