They still have to be accredited so yeah, actually, they do.
No. That is optional. They can be eligible and can be accredited if they want to. You can be accredited and still not have to answer to the government. That merely lets the students to be fully licensed/certified in whatever field they go into and show they have been tested by state/federal level.
What allows these universities/schools to not answer to the government is that they receive no funding from the government be it local, state, or federal they don't have to answer to the government on anything be it who they allow to come into the school, how they conduct the courses, and who they may hire and fire, suspend, etc.
Accreditation doesn't count as being funded by the government. Which are two completely different things.