Saint,
I think it will depend on how good the public schools are in your area. Some are awesome and some not so good. Check out
www.greatschools.com (or it may be greatschools.net, not sure) and you can look up the respective schools in your area and see how they have faired in national testing. The schools here in Deridder are actually pretty darn good and they are good in Kinder as well. I have been very satisfied with the teachers my kids have been involved with in the public schools. Just because a school is private doesn't make it a good school. Many private schools don't require any type of certification to teach. On the other hand, I think there are certainly a limited number of private schools that are a cut above strong public schools. These schools will invariably be very expensive.
I am a lawyer, my older brother is a physcian and my younger brother is a civil engineer and we are products of an average public school. Several of my friends went to private school and they thought their s****t didn't stink. Many of them were from wealthy families and they had no clue how to how to interact with blacks or rednecks when they graduated. Many of them had just about everything given to them growing up and they ended up being worthless as adults.
If your son in the sixth grade is having trouble where he is, maybe he is hanging with the wrong crowd. I would seriously consider a move for him, whether that be public or private, just to give him a change. Most times I think solid parenting can overcome a bad crowd, but there are times when a kid just gets wrapped up with the bad apples and there is nothing a parent can do except move him.
Best of luck.