Leftysidearmmom, that's kinda the way it works here. Home-schooled kids can play sports at their local public school. I don't know of many, but it's happened.
Also, private school kids can play at their local public school, if the private school doesn't offer that sport. Lacrosse, wrestling, field hockey... they were sports most private grade schools didn't have, until recently.
Then, there's the whole CYO thing. Sports are fifth through eighth grades. As long as you belong to the church, you can play for the church's school. Some churches (parishes) don't offer every sport, so they consolidate into one team, drawing players from three or four parishes, public and private schools. The results are some real powerhouses in a few sports, football, especially. Many kids choose CYO sports over sports at their own school.
The CYO doesn't belong to the state athletic association, but the Philadelphia Catholic League (high schools) joined a few years ago.