Interesting question. Two things come to my mind:
First, I think a lot of it is about self-presentation, and actually I think it's gendered (women tending on average to lack confidence, compared to equally competent men) as much as being a state/private thing. In some social environments (the private girls' school I attended, for example) it was the done thing to express a lack of self-confidence. If you said you weren't nervous about an exam or about giving a presentation, for example, people would think you were getting a bit above yourself. When people talk about "inde school confidence" I think they're often talking about other, often boys', schools where the opposite seems to be true - where the done thing is to express confidence in yourself, even if you don't feel it. This obviously affects how confident people think you are, but also, I suspect there's a bit of a self-fulfilling prophecy angle to this. If you get into the habit of expressing self-doubt because it is expected, you probably feel more self-doubt than if you're in the habit of expressing self-confidence because that is expected! Neither is the right course all the time, falling in with the expectations of the society around you isn't always bad, and you can't alter the social environment all alone, but I think as a parent you can help your child to notice things like this, and make a conscious choice about how to present themself, rather than just falling in with what's expected.
Second, for real confidence you need competence - that's so obvious as to sound patronising, but it applies to some "transferable skills" that seem to be more typical of what private schools do than what state schools do, although I'm sure there are exceptions both ways before anyone jumps on me. For example, my fairly shy DS doesn't worry about standing up and talking in front of a room full of people. He's had to do this frequently since he started at his prep school aged 4, as has every other child there. What I hear from my friends with children at our local state primary is that there are opportunities for children to do that, but they're not nearly as frequent and they're typically optional (e.g. there's a class assembly and some members of the class will speak, but not everyone has to). This is a small thing, but it might be the difference one day between being worried about the process of giving an important presentation to the board and only being worried about the content. And that one could be addressed by encouraging a child to take the optional opportunities, join the drama club, do debating, whatever.