I have had some women asking about the possibility of having a VBAC (vaginal birth after a caesarean) in a private hospital. "Is it possible?" they ask. Yes, it is possible, is the short answer.
How possible, is probably a better question, and the answer to this will depend more on the care provider you choose than the hospital, especially in a private hospital. In a private hospital, your care will be directed more by your care provider, and less by hospital policy. If there are specific things you are wanting or not wanting, the person to speak with is your care provider, and if necessary, they can / will write on your antenatal card so that when you come in in labour, the midwives will know what you and your care provider have agreed to.
All of this said, I always encourage women to look at the statistics of the hospital that they are considering birthing at. While your obstetrician might have a 50% VBAC success rate, the hospital itself might have a 3% VBAC success rate (I am not exaggerating here!). You might want to ask your care provider why s/he thinks this is so, and if there are any particular factors about that private hospital that make a VBAC less (or more) likely.