I don't think it revolves so much about being public vs private so much - more like hospital vs hospital.

For example. I have two hospitals near me. The closest one (1km down the road) is a huge public hospital - biggest one on the coast. The closest private hospital is 2km down the road. Best one on the coast.

Private hospital offers waterbirths - public does not.
Private hospital gets a single room with your own bathroom - Public hospital have to share PLUS you are not allowed to BF in that room - you have to walk down to a public sitting room and do it there.

For those two reasons right there - I will continue going to my private hospital. Not being I'm a snob, not because I want my baby delivered by a surgeon, but because that care suits me. I personally care whether or not I am allowed to experiment with BFing in private or a public situation.

But some public hospitals are like my private hospital. If the public one here wasn't so damn crappy and all they want is the bed empty - I would probably consider going there.

I say make your decision about the level of care that suits you from the actual hospitals that are near you - and don't let it enter your mind whether they are public or private.