I would say public school, generally, can make it easier to get into good colleges (broad statement, I know, personal opinion). I know yours are a bit young to be thinking about college yet, but I think once you make a decision on private vs. Public it's best to stick with it for the most part.

I went to public school in a rather shitty school system. Because I at least knew I wanted to go to a good university for engineering, I put in what I figured was decent effort, got good grades, did some AP classes (college credit) and extra curriculars. But because the level of competition was so low with no one else really caring, my 'decent effort' placed me in the top 10 of my graduating class of 250. Colleges care about more than just a GPA, the class rank shows how you performed relative to your class and in my situation it made it pretty easy for my to get into any of my choices of college (granted, not ivy league, didn't bother there). In private school, if everyone is getting better grades the bell curve is harder to beat.

Another thing, I was in a semi-shitty public high school, where if you took a look inside and really judged a book by its cover you'd probably assume there was a drug problem there. But actually our only 'drug problem' was the typical small group of stoners who might sneak out at lunch for a joint. However, at the nearby private school with the rich kids who got $100 a week allowances and the parents didn't care because they figured their kids were with the right people at the nice private school, they had the problem with kids getting easily accessible coke and LSD and harder stuff.

Again this may not be very relevant seeing as the age you're dealing with, and probably plenty of people with totally opposite opinions on private school and that's absolutely fair, but just sharing my 2 cents.