It is the law that anyone offering tuition should be registered with HMRC. Even if they don't earn enough to pay income tax, they still have to be registered. I don't understand why there should be any distinction between state school and independent school teachers. Many tutors are retired or taking a career break to bring up a family, so they're neither state nor independent school teachers.
I'm not registered for VAT, because I don't earn anywhere near enough and it's unlikely that I ever will.
I have a pack I give to parents when I start tutoring, which includes copies of my certificates and DBS certificate.
I expect there are tutors who aren't registered or even qualified, but that's up to parents to check. I know there are some who try to get tutoring on the cheap and aren't too bothered about qualifications. Sometimes university students do tutoring for extra money, which is common in other countries.
I know there have been calls for all tutors to be registered, as childminders are. It sounds like a good idea, but I have reservations. Firstly, I doubt if unregistered tutoring can be stopped. How is anybody going to know about the neighbour taking a few pounds for doing a favour?
Secondly, the people behind the calls for registration are a couple of the big players in the private tutoring business, who want to push out independent tutors. Typically, they charge about double my fee and don't actually do the tutoring themselves. Their main selling point is that they've checked tutors' credentials, but any parent could do that. I don't know how much registration fees would be, but a figure of about £250pa was being thrown around. For any teacher only doing a couple of hours tuition a week, that's a high percentage of profit. Even for me, I would have to increase my fees to cover the loss.
This has nothing to do with paying tax. As with all self-employed, there are probably those who do cash in hand work. To be honest, I don't really understand the point of raising the issue in this context.