PRIVATE VS. PUBLIC

There's no comparison between public and private school almost - which is Not to say one is better than the other. Both are schools but in many ways, it's like comparing apples and oranges.
They're that different. Not all private schools have smaller classrooms but many do. Not all private school have large lush rolling acres of campus but many do. Not all private schools have articulated philosophies of education but many do and many also have long-standing traditions and very defined goals of the kind of people they want their students to be. (whether they achieve that or not is another question of course)
Each private school is different - few generalizations apply including the one that the differences between private schools and your local public school are worth the money that private schools charge - it's a Lot. I had my own children in public school for a while and in private school and taught in both too.
Your second question stumps me - in my state I know of no private school that has requested to take the state tests - in fact, private schools usually pride themselves on not having to do such things. Some parents seek out private schools because they operate more independently of government requirements such as state tests.
Many private schools though will have their students taking other standardized tests - the ERBs are a popular choice of standardized test for the private schools.
If though you're asking - can your children in private school go to the public school to sit for the state tests - that's an intriguing question. I'd follow it with - what would be the point? State tests do Not give individual scores -they give only aggregate data. If you're thinking, you'll get your children's scores on those state tests, I don't think you can.