I work in a private school in France. When they employed me they knew that I had next to no experience of teaching, I had not worked with children before, they never asked for any of my certificates and they never did a police check. I take classes of 14-15yos on my own, but have also taken classes of 8-9yos. I also had an interview in a private primary school, again I had to ask them to photocopy my certificates and my Police Check certificate. They offered me the job taking classes of 8-9yos by myself but I couldn't take it due to lack of funding.
As far as I am aware, the system for employing teachers in private schools in the UK is much the same. Compare this with when I applied to be a teaching assistant at a state primary in the UK, they couldn't do enough checks on me! Plus the interviews for teachers are pretty rigorous, you need a high standard of education in pretty much everything, not just the topic that you'll be teaching.
So based on that, I'd vote for state!