I would expect the public charity hospital, Grady, to get most of the criminal business, not the private Methodist hospital.
I don't like to reveal too much about myself on the web but I'll make an exception:
If Atlanta is anything like Boston (where I worked) Grady almost certainly got some,if not many,prisoners as patients.That was certainly the case in Boston...Boston City Hospital (renamed "Boston Medical Center" some years ago) certainly got prisoners as patients,particularly from the Suffolk County lockup...only a couple of miles away.But,as I noted,my hospital...a very "private" one...also received such prisoners,including from Walpole State Prison (AKA: "Cedar Junction"),the home of the state's nastiest criminals.
It seems to me that the location of the state's large prisons could be a factor.Walpole State Prison is located within easy commuting distance of Boston.Georgia is a much bigger state so the same might not be true there.
But one thing I can guarantee is that regardless of distances involved hospitals like Emory would,at the very least,have an agreement with the Georgia Department of Corrections to provide care for the sickest,most "complicated",patients.