I grew up in a small community and attended the public schools. The nearest private school was a 30 mile drive.
My high school had about 200 students, my graduating class had about 52. Average class size was about 17. We co-opped with a neighboring school that tad smaller in all but one sport (basketball). This co-op gave us some extra $$ for sports and a better pool of athletes.
As far as the education goes, I feel that it was excellent. When I went into college, I felt more than prepared for the classes that I took. Then again, as for me, as well as many of us, it is hard to judge the quality of the school, as opposed to the quality of the student. I graduated sixth in my class and got the most that I could out of my high school education. The smaller school made it easier for me to participate in sports that I wouldn't have been able to make the cut for in the larger schools. The small size also made it very easy for a teacher to go one on one with a student if he/she was having troubles.
I don't feel that private schools are better than public in general, but in the extreme cases, I believe that there may be situations where the private solution may be the better. Two of these extremes that I can think of are incredibly small public schools, and oversized, overcrowded, public schools.
When a school is too small, it will not have the resources both financially, and personel wise to provide a student with enough choices.
When a school is too big, it makes for too many choices and often lacks the refinement that a smaller situation can provide. A larger student body makes it harder for a teacher to see students with learning problems and in turn, making it easier for students to pass that probably shouldn't. Another problem with large schools is sports. Many large schools make it next to impossible for a student who has the desire, but not the talent to participate in sports. In a smaller environment, this is not a problem.
In my opinion, private schools are an option, but not a solution. Many of the private schools around my area are catholic/religious based. I would never subject my child to that...but that's another thread

From my experience, the catholic high schools generally had higher drug/alchol abuse rates than the public schools. But then, this is just my experience with working with several private schooled students from various private schools throughout my high school years.
And, as many have stated, it's the parents - especially the mothers, that provide the educational encouragement that a child needs to excel in schools. Multiple studies have shown that if a child's mother expresses interest in her child's education, the child will do better in school than if the father, or in worse case, no parent, shows support. This alone, can be the determining factor in how effective a school is.
I won't say that I would never send my children to a private school, but I will say that it be highly more likely that I would pay the tuition difference to have my child go to a better public school in my area first.