I believe a child's family life has much more to do with their confidence than their school life. They will model your behaviour regarding how to talk to people and how to behave in social situations. My children's social skills are in no way lacking compared to their privately educated cousins.
Our dcs are at a state school, they are encouraged to speak out in class and they have many school based extracurricular activities which encourage confidence to develop. Some people on here don't seem to realise that some state schools have debating, drama, chess, orchestras, choirs, sport, pools, fantastic science facilities, foreign trips, yachting, skiing...
The problem is that provision is so patchy, we benefit as there is no grammar system and no decent private schools locally, so everyone sends their dcs to the comp, regardless of income, educational background or whatever. For example there are many families here where both parents have PhD level education, and are earning £150+ between them, and also many who are on average wages and benefits.
The academic results will never match a private school, because of the non selective intake. Our head said he was equally proud of the profoundly autistic child who has just mastered writing his own name, as he was of the student who had just got into Oxford, having discussed philosophy in Mandarin at her interview.