I went to a private school, but I didn't pay any of the fees because I was on a couple of scholarships and the remainder of the fees were paid by bursaries so even though I went to a private school, my background was emphasised (on my reference) as neither of my parents went to university properly (apart from my mum who finished a degree at our local, crap, uni a few years ago) and we have a low income. We have a Director of Studies who overlooks all of the school's results and teaches Physics and he does all the Oxbridge forms (the extra ones) and explained the application process to us and asked us how my interviews went when I saw him. I think it varies a lot though because some private schools offer loads of coaching and push people a lot