It totally depends on the school. I don't know what neighborhood you live in or what the public schools there are like, but here are some things to consider:
1) Is quality of education, class size, etc. your only consideration? That is, you aren't considering sending your kid to private school for religious reasons or something like that?
2)Other moms in the area are your BEST resource. Ask for the number of the moms on the PTO/ parents' group at both your local public school AND any private schools you're considering. Ask them frankly what they and their children like about the schools and what they DON'T like or would like to see changed. But- be aware that in my experience a lot of private school parents REALLY want you to join up and are a little more 'cheerleader' than public school parents.
Remember that you don't have to become best friends with other parents at your kid's school- but realistically, you will be interacting and fund-raising and room mothering with these other parents for several years to come. Do you get a friendly vibe from them? Do they and/or their kids seem nice to you? Can you see yourself chatting with them about school, waiting for the kids, getting coffee, etc? All this stuff WILL come up in the future, so just think about it!
3)Call the schools you are considering and ask for a tour. Meet with the principal and, if possible, the pre-school or kindergarten teachers (their schedules may be busy though) Try to get a 'feel' for each place- is this an environment where you can picture your child busy and learning and making friends? What does the social atmosphere seem to be? What is each school's philosophy?
4) Ask to see the school's test scores and reading and math averages. Usually they start testing at third grade, but look at scores all the way through middle school if possible. See if they remain consistent throughout the grades- remember, your child may potentially be here for several years, not just kindergarten. Another good resource for comparing schools and seeing if people have commented on them is www.GreatSchools.com - a website that many schools post test scores on and parents comment on. Check out the schools you're looking at there.
5)Convenience. Where is each school located? What is their schedule and how does it fit with your own. Again, remember, if this school works for your child, you will not just be picking up a little one- someday in just a few years he may be walking to school or riding his bike with friends. What do you think of the area the schools are in and how close are they to you? If you go back to work in a year or so, will the school still work for your schedule- these are just things to keep in mind.
My son attended a private daycare- very small and personalized- but has always gone to public school. In kindergarten through 1st grade, he was at a very diverse school which exposed him to lots of kids from different ethnic and financial backgrounds. He had a GREAT experience there, because of the excellent teachers and administration, even though the school itself had financial and social challenges.
We bought a house when he was going into 2nd grade in a different school district that has an award-winning public elementary school. Although it is a public school, the largest class he has ever had has been 20 kids- last year in third grade, his class had 15 kids in it- and you can't beat that!
Again, it is the quality of the teaching and the fact that it is a smaller school that seems to make a difference. My son is in 4th grade now and reading at a high high-school level and in the gifted program for both English and mathematics. We have a school organic community garden, art, gym, etc.
The cost of private school is prohibitive for us anyway, but even my fiance, who went to private schools his entire life, (Montessori and a prestigious Catholic Prep academy,) says that he would absolutely not want to send his kids to private school and that he really thinks public school- if the quality of education is good- is a much broader and better experience for kids.
As for the myth that kids at private school are somehow better socially or a better 'influence' etc. Frankly, that is bull. There are some bad apples in every barrel- plenty of people with the money to send kids to private school have kids who get into trouble or have bad habits,etc. You need good involved parents, and good teachers.
Make the best choice for you, but do your research! Good luck and have fun as a school age mom!