I went private for surgery and chemo because, like @romla, I had had years of private insurance. I went public for rads and had no complaints - I would have paid a fortune for private rads and as far as I can tell, there would have been no difference in treatment or care except that maybe because my public hospital followed the latest treatment protocols, my treatment was shorter than it would have been likely in private. As for @romla, follow up care from the private hospital where I had surgery was poor. However, my cousin went for public for surgery and the waiting list was a few months whereas I waited 2 weeks (and it may have been sooner if I hadn't asked the surgeon to have that extra week). The other benefit of private is that with my cover I can get a lot back on mastectomy garments and it has covered a number of weeks of oncology rehab.
If you go down that path, make sure that you get very clear advise on waiting times having had bc (be specific and I'd go as far as to ask for it in writing) and also compare what the different levels in the tables cover.