I went public with my daughter 4 years ago. I had to attend the hospital for all my app's and there was usually 3 others booked in at the same time. I did have my own room after the birth, but apparently this was because I'd had an Emergency C.
The only out of pocket expenses I had were for my scans (difference between there charge and medicare rebate) and that was it.

I went private the second time as I'd had alot of issues getting pregnant and I wanted to use the specialist I had been seeing the whole time.
My fee to see the Specialist was $3500. $2000 at 20 weeks and $1500 at 30 weeks. I then had to pay $350 excess for private hospital. I paid for the scans myself and claimed back the money later.
After paying the $2000 we were able to claim back 80% after the initial medicare gap was paid. after that we got back 80% of all out of pocket expenses except for the $1500 we paid at 30 weeks. (we claimed this at tax time and got about $150 back).

After paying everything, we ended up worse off going private, but if I have another baby, I would like to go Private again. I like the doctor I used and I also had a Caesar the second time.

Hope this helps with your choice