Extremely subjective question. Both public and private schools can hit the low or high end of education. There are stellar examples of each and horrible examples of each.
My sister put my niece in a private Christian school when she was younger. She eventually pulled her out because the school had a policy of letting the upperclassmen be in charge of the younger students during breaks and basically ignored any complaints of bullying. Then, after my sister put my niece into public school, she learned that she was way behind in math and a few other subjects. She came very, very close to having to repeat a grade because of that. It had nothing to do with ability, either, as my niece would make the principle's list the next year.
Every school should be evaluated independently. It should never be assumed that being private means the school is superior.
With homeschooling, it comes down to whether or not the teacher has the ability to teach everything the child should know, and if the child gets enough interaction with other children otherwise. Learning how to fit into society is an important part of school.
