As someone who has taught at two private schools, I would say it varies both by pupil and by school.
I would say my first school (Central London, girls, religious) was much happier than my second (outer London, mixed, CofE but hardly). The main difference, I would say, is that my first school genuinely did value every pupil as an individual, and this permeated down through the school. My second, although improving, was headed by an ogress for many years (the current head is far better) but it takes years to turn around a school, as a culture (in this case, perfect behaviour and success),gets embedded in the whole staff body.
But even in my first school, there were a few unhappy pupils; perfectionism and anorexia were problems and some pupils just didn’t fit in.
I do think, to some extent, it is better to ask teachers than parents, as many parents won’t really know and, particularly when they have invested large sums of money, won’t admit when they have got it wrong.
My children go to a good local comprehensive and I think that they are both happy. I think one big difference is that school, for better or for worse, is a much bigger part of your life in private-longer days, early morning and late evening clubs and sports coaching etc etc.
You should go to lots of schools from both sectors and look around. If you care about happiness, ask those giving the tour and the head lots of difficult questions about it and avoid exam factories, even if it means possibly dropping a grade on average.