Private schools are afraid of lawsuits, too, obviously. They protect themselves by:
1) Strict standards of behavior that are enforced. If a kid screws around too much in a private school, they're out. Further, private schools limit who is admitted in the first place. There's normally a smaller staff/student ratio at private schools and students are monitored more closely. I've done some work at private schools and I wouldn't want to be a student there despite the prestige. 2) Contractual limited liability: Unlike a public school which is governed by law, private schools can and do have contracts with students limiting the schools' liability. (Public schools do that for some optional activities, like school sports).It does at times, although the law has granted some immunity to schools and coaches if a kid gets injured.
Liability law is a tricky thing. It is often set by the courts by granting a lawsuit over something. However, legislators may limit liability in certain situations and they do on occassion. Often these issues become very contentious (ie limiting malpractice awards, for example). They generally protect sports coaches because no one would host a sport if they'd get sued every time a kid got injured.