I went to both public and private school and in my case I did much better in the private school but I lived in NY city where there are many many different types of private schools, there weren't choices for public schools other than specialized high schools when I was going to school.

I went to a progressive school, which was a great fit for me. It wouldn't be a great fit for every kid. I do agree that it does depend on the kid, some kids will do well where ever they go and others need a specific kind of environment to thrive. There isn't an easy answer as I don't think private school is always the answer especially if there isn't much variety, yes the academics might be better but the pace, culture etc may not be right for your child or your family.

I don't think there is a right or wrong answer. The only major drawback to public school for me is the standardized testing and not so good teachers and/or schools may focus on "passing" the test as opposed to actually teaching. For me that is a concern. However, magnet or charter schools may not have the same requirement to test every year.