I've only worked at one private school and two state schools, so I haven't got that much advice, but I would agree with puffin that it's swings and roundabouts. Finding a supportive and positive school to be in, regardless of sector, is always my priority. I had a bad experience at the private school because of some parents who constantly had something to criticize about my lessons/homework. The head always sided with them and the situation made my life a misery. In a state school now and happy as can be. Head is lovely and I'm valued and supported every step of the way.
Not the most helpful of contributions, but that's my two cents.
I teach in the private sector and have to say it is generally great. (I taught in a Primary School before that for many years).
In my experience the teachers are trusted and respected more and are allowed to get on with teaching. There can be ups and downs like any school. Parents can tend to be much pushier as they are paying for education. Expectations are much higher. It's swings and roundabouts and also very much depends on individual schools.
However I loved teaching in a Primary School and do miss it!