Every authority, every jurisdiction should have authority extending only to its borders with limited exceptions (embassies, field trips)
If it's something serious then there will be a law about it and the law can be enforced against the student. If less serious this is something the parents should be handling.
Schools should only be concerned with what happens at school. We have to remember that Public Schools are government agents. Allowing the state access to the home is an invasion of privacy and a deprivation of rights. It is literally creating a "nanny state." Leave the schools in the schools!
It is not the responsibility of the school to punish children outside of the school grounds. This should be left up to the parents. Schools do not and should not have the right to punish children during a time when they are under the supervision of their parents. Once a child leaves school grounds, they are the responsibility of parents and it should stay that way.
Schools should only hold students accountable for what is on school grounds or what has happened under the supervision of faculty or staff. What happens off campus is none of a school's business. That's the business of a child's parents or legal gaurdians, and it needs to stay that way.
When a child misbehaves in school the teacher has every right to enforce the rules on the child's misconduct and punish the child to how they see fit. Part of the reason a child should be punished in the school environment is to set an example for the other students, indicating a clear line that should not be crossed or else. Teachers of young students for better or worse must also be disciplinarians along with teaching. However, I do not believe this power should reach beyond the school grounds, that should be reserved for parents or if serious crimes are committed, police. If a teacher gets wind of mischievous behavior outside of school grounds they should inform the proper authority and let them deal with the situation, not deal with it themselves.