Babyluv - I'd do two things. Have a look at the stats obviously and if you do decide to go private, ask the ob for their caesar rate - that would be a pretty good indication of how pro-natural birth they are. It's not just about the caesar rate obviously, there's also interventions such as inductions, episiotomies etc. but I think their caesar rate would be a pretty good indicator.

Then compare that with the public hospitals. As someone else said, when it comes to private hospitals there could be a big variation between doctors. For instance, my ob has the lowest caesar rate in Melbourne but the hospitals he delivers at still have quite high caesar rates (I think one of them may even be as high as 50% but it's a while since I looked - that's going on what someone told me recently). If his is quite low (so say 20% which is not that low but just for the sake of argument) and the hospitals is 50%, there must be obs delivering in that hospital whose caesar rate is much higher than 50%.

I imagine you wouldn't see a big difference in caesar rates between obs at the same public hospital but I could be wrong and I'm sure someone else will pop in with more knowledge about that.