I think an amalgamation of the available canon evidence would suggest that indeed, on core Federation worlds like Earth and Betazed at least, Federation citizens are given public schooling and have their medical needs provided for.
I believe the only teachers we have seen were volunteers: there was no mention of Keiko charging for school on DS9, and the teachers on the Enterprise were not part of her crew. We have seen many students being "home schooled", and others making reference to "school" but no indication of who runs it. You might infer that the schools are run by the government, but I think there is at least an equal amount of evidence that the schools are run by the citizens themselves, teachers being either volunteers or, if paid, paid by the parents of the students directly.
We have seen Starfleet providing medical care, but we have also seen doctors who were not part of Starfleet. We have no indication if they work for the government, for some private entity, or if they provide medical care as volunteers, simply because they are able to and see the need. If anything, that last has a bit more canon support than the others.
That their medical needs are provided for does not mean that they are provided for by the government