personally i think it depends on a mixture of factors whether your child does well at school or not but at the very least half of how well you do in life is down to YOU, the child at school.
the proof of this is in that you get loads of intelligent, amazing kids from both state school and private.
where i live we have 2 well known private schools and 3 state schools in close proximity. the private school kids [boys only] are all notorious for being excessive drug users, abusive to staff, and have awful, awful attitudes towards women. [my friend is a teacher there and she would leave if she had another job to go to in a heartbeat]
the girls school, from friends of my daughters who i have met, appear to be mostly concerned with having as much time off as possible to enjoy the ski season.
considering the lovely, well rounded kids [with the few bad apples that are at every school, private or state] that have come out of my daughters state school, it would be easy to make judgements.
but i dont. because i can see how private school can greatly benefit those who are depserately in need of those smaller classes and 1-to-1 attention [my friends DS who has autism, for instance, although he goes to one out of town]
whenever people go on about private schools being better i feel sad for them that they havnt been lucky enough to live near decent state schools. league tables mean nothing, its your individual childs grades you need to be worrying about.