What / Where was the first public jail that wasn't some kings personal dungeon? And how did the modern idea of a police force that serves the public come about? (they weren't personal guards or soldiers anymore.)
This question is badly phrased which is why you're probably not going to get a full answer. The concepts of prisons and police forces are modern which, while technically post-Roman, are more than 1300 years 'post-Roman'. As post-Roman is in your title most modernists will skip over it and most post-Roman/early medievalists won't be able to answer it - at least in any level of depth.
I'd recommend resubmitting with a different title. Also, have a look at this thread.