It kind of depends on your reasons for going private, and the hospital you choose.
Private hospitals tend to have much higher c/s rates than public, so if thats one of the parts you want to avoid you will also need to research your OB really well.
You can still have room changes in private hospitals - my gf has changed rooms with her private hospital, in one case twice, in both pregnancies.

You should be guaranteed a private room though if you go private.

I cant help with the distance thing - how long was your labour with your first bub? If it was really short then Im not sure how far away I would travel.

Oh and for me, distance keepsing visitors away would be a bonus.