I know people who have taught in a top private ("public") school for many years. From what they tell me, many teachers aspire to teach in the private sector, because the class sizes are small (10-12 max), the facilities are good, and often the pupils' parents will be very motivated to push their children to achieve good results, as the fees aren't cheap so the parents don't want to waste money if their kids won't put the work in. Plus, the wages are better.
So, for all the above reasons and many more besides, teachers rarely move from the private sector back to state schools, and truth be told, after years of teaching small classes in well equipped schools, chances are they'd find it pretty challenging teaching classes of 40, with poor facilities and violent pupils in the mix (like the awful school I went to).
I feel sad about the unfairness of some kids having a great private education while others receive a poor education in overcrowded, underfunded state schools, but I don't think abolition of private schools would solve that. I don't agree with private schools having charitable status though, as many of the parents are absolutely minted and can well afford the fees. A private education opens doors and gives you advantages that others don't have, and the "old boys' network" looks after its own regarding top jobs. Charity is the last thing these fortunate families need, as the private school system perpetuates the status quo, where wealth buys the best education, the right accent and decent contacts, which often leads to lucrative jobs. Even so, I don't think abolition is the answer, but I'd like to see better standards for all children.