This is a broad topic, which you can take in many directions. So, you need to be selective. In other words, you need a good thesis statement that will drive your essay.
Here are some examples:
"Even though private school offer excellent education, they do not offer a realistic view of the world." This can be a good and cogent thesis. You could mention the costs associated with private education, which necessarily limits people. Moreover, society, unlike a private school, is does not only interact with a certain group of people. Therefore, learning to interact with all sorts of people is a valuable lesson at all ages.
"All education should be public, because we should create a society where all people have equal access to education." This is a more difficult thesis to defend, but you might mention things like social inequality, the advantages of social capital, and anything else that gives the wealthy a leg up.
These two statements should get you started.
Your paper could center around the pros and cons of each type of school setting. I would discuss what each type of school has to offer its students. I would then compare and contrast the graduation rates of each type of school.