I've experience of both sectors, and sure you get some terrible duffer teachers at private schools, but on balance, broadly speaking, I'd say the teachers were better at private schools.
I suppose it depends on how you define 'better' and how you view a 'better' teacher?
IMO a teacher must ideally be educated to a high standard themselves, that's a prerequisite, before you look at anything else. They should have a firm grip of the basics, good grammar etc. They should also be articulate, intelligent and be able to control a class. Something else that is important, for me anyway, is HUGE enthusiasm for the job, absolute passion and a love for children.
Good private schools have a rigorous selection procedure, the Dragon school in Oxford, for example, even invites teachers to have lunch with the children before they are offered a job. They don't know they are being observed. Those that don't interact well with the children don't get offered a job.
For me the biggest factor in saying private schools are 'better' though refers to the class sizes. My son's teacher simply doesn't know him and what he's capable of, she can't, the class is too large. Mistakes have been made as a result.
My sister has a daughter at a prep, she's one of 15, she's not underestimated she has a dialogue with the teacher and she's listened too. Not that this wouldn't happen at a state school but there are so many other demands on the teacher's time they simply can't know the child as well.