I believe in finding the best fit - the most suitable learning environment - for a child.
We've tried private school. I liked the academic focus, the extra communication with parents (when parents are paying, the school seems to want to show what you are paying for), and some creative educational opportunities that were offered.
I didn't like paying large additional levies for building funds, grounds maintenance, and facility fees - labs, sports equipment, library, computers, etc. in addition to the annual tuition. I didn't like the expectation that we would also fundraise for these things, in addition to paying the additional levies. It seemed like there was a constant hand reaching into my pocket. Many people complain about paying extra fees and fundraising in public schools, but IME, it's much worse in private schools. I was also unimpressed by the atmosphere of entitlement that permeated the student body and the parents - clothes, jewelry, iPods and game systems, expensive vacations, joining private clubs and dining in expensive restaurants....
IME, private schools market themselves as offering academically enriched programs. They tend to squeeze out the students with learning disabilities because they can't keep up. By the same token, they are less likely to offer truly specialized gifted programs, because every parent thinks their child is academically gifted - so they all want any advanced or enriched classes available to their children, whether they would be identified gifted or not. While the academic focus was more intense than public school, I found that it didn't serve specialized populations at either end of the spectrum.
Having said that, though, there are some good private schools in our area that have a niche market for special learning needs. There are a couple of private gifted schools, and several schools for learning disabled students. I think the key is finding the right placement for the student.
We've been happy with the public schools. There are always things that could be improved - communication with parents varies. That's actually getting better with e-mail and website updates. The good experiences outweigh the bad though. My dc have attended gifted programs with teachers with special training and lots of experience with this population. They've had access to interesting learning opportunities. Their fellow students are diverse - both in learning styles and culturally. The schools have been open to parental input - possibly because there seems to be more accountability and oversight - we can take issues up with school board supervisors and/or elected trustees if we aren't satisfied with a school's response to a problem. Private schools with wait lists seem to be less responsive - you can leave and pull your money, but they have another family to replace you if you decide to go.