Never, ever assume private covers means you get your own room!
Okay, IMO, about the best part of going private would be the high chance you'll have your own room. Thats about the only positive thing I can think of about going private It certainly influenced my decision first time around. If your own room is important to you check with the private hospital you intend to go to and they can verify it your own room is guaranteed or not. In the hospital I went to there were only single rooms in maternity.
This time around, my own room isn't so important