Let's start off with homeschooling. So parents want to homeschool their children. They get the necessary certification etc. and off they go.
Now, we have a certain number of parents in a neighborhood who are doing this. They figure out that they're duplicating a whole lot of effort and decide to pool their resources. They also think their kids could benefit from interaction, and so they find a place, maybe one of the parents' homes, to school all the children together (this is actually illegal in NC, and probably many other states, too; so this isn't even allowed to go this far).
They start inviting other parents in, who join. They also go around to businesses and others seeking donations for the school. Pretty soon, they're able to take in children even if the parents themselves aren't paying directly. The more active parents become a board, and they may even raise enough for a building of their own.
Continue this trend, and you end up with a public school, voluntarily funded, no force required.