We homeschooled for years. Now our kids are in public school. Both homeschooling and public/private schooling have plenty of good outcomes.

But there are some positive peer-induced socializations. Both my boys had some quite annoying habits of speech and mannerisms that are gone now. I think part of the reason is that their peers are less understanding and forgiving of such behavior than their parents. Similarly, they are more receptive to peer criticism, as it packs more of a threat. Other kids will ostracize you; parents, not so much.