Most of the rooms at the hospital I stayed in are private. Ask your doctor's nurse about it, he/she should know. You also might ask when you go for pre-admissions paperwork. I think you can request it (if your insurance covers it). If you are paying out of pocket, then ask for what you can afford and think is reasonable.
Good luck with surgery! I had mine 2 years ago yesterday. I had an LAVH.
I will call the hospital and find out more. I know that my insurance covers a semi-private room and if I wanted it I would just have to pay the difference for the private room. Still deciding on whether it would be necessary.