We also have both private and public schools but charters school are a mix of both. They are private but funded publicly. And they are not under the same public scrutiny, regulations, laws, etc... For all that, they fall under the private sector side of things. So they get public school money but private school operations.
Yeah, such a thing is never a good idea, usually it just combines the bad things of both systems. And such a thing is often used to channel public money into private hands without much regulations.