It depends on what you mean, to give birth privately you will need to be an established patient of the obs. They generally won't just delivery your baby on the day, they would also need/want to do your post and prenatal care. Most do payment plans but you pay a birth fee plus your normal consultation fees. When I had my last in 2012, I paid $2200 as a birth fee (I was in labour 37 minutes so I felt a bit jipped -but you never know), then $75 per consultation. I got around $440 something back from Medicare, then $32 per consultation until I hit my safety net then I got most of it back. If you need cs in an emergency you would also need to pay for an anaesthetist, I have no idea how much, but probably thousands, my DH was out of pocket $400 last year and we are fully insured.
The cost of my local private hospital is $1450 a day plus theatre fees.
Did your obs indicate they would attend your delivery but not do pre/post care? I've never heard of that, but if you are low risk they might.
I was chatting to a friend who had her baby last year by elective cs and she was out of pocket $4200 in total with the out of pockets on doctors + her hospital excess. I would add around 60% to that if you are going the full hog. Of course, this is in my local town where obs are in short supply and there is only one private hospital so it might be more competitive in different states.