DD goes to a very ordinary primary school, though she will be going to a private school in September (one of the ones people have probably heard of, very selective and very academic). She is hugely confident, articulate, able to present her own opinions and back them up, amazingly focused on her education and apologises to nobody for things like her academic bent and desire to know things and find things out (though there is a certain amount of ridicule of this, even at primary level).
Her primary school does quite a lot of the things that people from private schools have mentioned upthread - opportunities to eg show adults round the school (when they were interviewing for a deputy head it was the children including DD who showed him round, and DD has been doing tours of the school for prospective parents since Y4 - the only school in the local area that does this rather than staff conducting tours), opportunities to debate and speak in public from the earliest years, opportunities to interact positively with younger children (reading buddies, maths buddies, opportunities to interact with adults in positions of authority (governors' teas, headteacher's teas etc, though they don't quite run to dinner parties). So I think her school has given her a lot of space to start to become an adult and take on responsibility. It's not a high performing school but it is a school that places a lot of emphasis on the next step and preparing children for independence. Interestingly, because it isn't a high performing school it isn't popular. But I think they are doing some things very right. Even though DD's class has a massively wide range of ability, they are all being taught to conduct themselves well in the world and I think it will stand all of them in good stead. I think the key here is that children aren't just children, they are people who will become adults and they need to practise this.
As well as this, I went to the same high-flying school that I'm sending her to. I do have an expectation of being able to negotiate in order to arrange things to suit me. I do have an expectation that things should, generally, be arranged for my benefit and the ability to follow through to make sure this happens, I do think that I'm capable of anything I want to do, and I have benefited from the sorts of opportunities to excel and to take responsibility that private schools are good at offering. I think I have probably passed this on to some extent. I was always treated like an adult at school and expected to behave in an adult way in terms of being responsible and thinking things through.
We didn't have particularly tiny class sizes or the same tutor all the way through (I think this is an awful idea, what if you don't get on with them?) but we were encouraged to believe that the world was ours and we could do whatever we wanted with it - and I've tried to pass this sense of possibility on to my daughter. Obviously, if you take this to extremes you get Boris Johnson or David Cameron and that is not good so perhaps don't emulate Eton. But it is important for children to believe that there are no limits to their potential achievement. After we took our O Levels, we were offered work experience. The offers were things like GP surgery, advertising agency, Lloyds of London, publishing company, law firm etc - all interesting careers that anyone would be fortunate to enter if they were interested in that area. My friend from university who went to a comprehensive in a deprived area got given a week at an abattoir.
I don't really have any connections as such and that's not what I hope DD gains from her future private education. I just want her to believe that nothing is off limits and she can do whatever she wants with her life as long as she tries hard enough.