Hey Sara - great question!
I think it is a bit different between the public and the private sections, I just went and saw my local GP - she searched her 'database' and selected a few surgeons that she knew of that were good, then we picked out a couple that specialised in rectal surgery - then from there I picked the one that was located closest to my work as I knew I'd have to make a couple of trips there.
I had the option of going private or public - I chose public hospital because the wait was really not much difference and I was assured that my surgeon would still do the surgery.
As it turned out, my surgeon was 'just there' and did not actually perform the surgery, but luckilly for me the lady who did, was wonderful.
If I could do it again, I think I would go private - there are more choices with surgeons, and I think the after care would have been better.. I was pretty much thrown out the door still drowsy
I'm sure the health systems in each country work differently - so make sure you do your research well before booking anything!!
Hope this helps