I agree on home schooling having a socialization problem. Private schooling is kind of the opposite: the most measurable effect of private schooling are the connections formed, followed by similar effects of having a more wealthy demographic.

I was homeschooled for most of my pre-univeristy education. I disagree with the notion that homeschoolers have a socialization problem. In my mostly rural Ohio county of about 50,000 people, there were around 50 families homeschooling about 100 kids. There were regular events like drama and chess clubs. I never felt like I didn't get enough socialization.