My twins attend public school up until I married my wife, in which my father-in-law decided he would pay the tuition for them and any furture children to attend the non-religious private school. I toured the school first, read the the curriculum, etc, but we took a year to decide if it was bes for them. The kids transferred there when they were 7, and the specific school has great academic and arts programs, small class sizes and a strongly enforced zero-tolerance policy. The public school they attended at first wasn't that different in terms of curriculum and safety, but the private school offered a more personal approach to education that I really like.
The private school had better test scores and that sort of thing, but I think that is skewed because of the initial size difference of the schools.
My kindergartener enrolled at the same private school this year and he's pretty happy there and I like how he's progressing socially and educationally.
I've always gone to public schools, though, but didn't attend them in my current country until I was 14. In my native country, the school system is extremely different! I received a great education, but my graduating was very large, so unless you specifically asked teachers to keep you motivated, the school seemed pretty dependent on students having a lot of self-determination (which isn't too easy for teenagers lol).
Every public AND private school is different here, though, depending on the area of the city they are in (we have school districts, which make initial decisions, and then the OPS that rule it all). Actually, I think that's why I like private schools, because they have the ability to be more individualistic and not all of them take the "every kid should be exactly the same" approach.
If you can, you should tour several of both types of schools (I don't know if its like here and if you want to register with the public school, you have to register at the school in your district. It was not like this in Tokyo, so it still confuses me haha). Meet teachers, principals, get curriculums, check out the facilities, and other things that may be important to you (like, for us, the quality of the lunch program was very important.. we did not like the OPS choices in that regard). If you can, you might be able to sit in on a class if you feel so inclined.
Don't get too nervous, though. I think both types of schools have pluses and minuses, you just have to decide which would suit your families' specific needs better.