Most charter school operators make a big deal about how they are publicly funded. In fact, it’s in their name. However, quite a few groups of teachers at such schools in New York, California, Chicago, and Philadelphia have begun forming unions. According to this article, some of the charter school administrators say they aren’t public after all: they are ‘private entities’, so they have the right to deny their employees the basic democratic right to unionize.
BTW, the membership of the Washington Teachers’ Union has voted a couple of times to authorize teacher-union membership drives at the charter schools. As far as I know, nothing much has happened.