A teacher should not be allowed to search a student's personal computer that is based at home. If, however, the student has a computer at school then I believe a teacher has every right to search it. When a student is on school property that makes everything they bring with them subject to searches, including their computer. I honestly don't think the teacher would find anything, most students are too smart to leave something readily available but I do feel the teacher does have the right to search it if they see a need.
If it the child's property then the teacher should not be able to search it because the teacher does not own it. If the computer is school property the teacher can search it because it is the school's property. Please people, start trusting kids more. The more you trust them, the more trustworthy they will be
If we are referring to personal property, the no, teachers should not be allowed to search them. Problems or concerns with a students personal property should be brought up with the parents or simply brought to the parents attention. What a student has on their on personal property, is not something a teacher should be privy to and it would also be a breach of the students privacy.