In the present day, drugs are a major problem in public schools. They probably have a presence in private schools, also, but I never attended one and so do not have any experience in that department. Drug testing, while invasive, could solve many problems at schools. It could reduce violence (caused by students under the influence), help remove drugs' presence from schools (if nobody can have it in their system, how can they take them and still show up to school?) and help communities.
A few years back, a seventh grade student from the local middle school broke into a skate shop and the local grocery store. They stole some shoes from the skate shop and were attempting to take alcohol and cigarettes from the grocery when they were apprehended. All of them had backpacks with them, and all of the backpacks had drugs and drug paraphernalia. It was discovered later that all of them had drugs in their system when they broke into the stores.
Drug testing wouldn't take much time out of students' lives, either. If all you have to do is urinate into a cup, why would you complain? You already have to have vision, hearing, and scoliosis testing. It would just add an extra 5 minutes to the health testing. The results could be sent home via snail mail, or, for added security, emailed to the parents. If you're wondering, "How can e-mail be more secure?", it would prevent the student from either disposing of the results or tampering with them.
Drug testing is already a part of big-league sports and some colleges, why not secondary schools as well?