Actually the ministry has a trace spell, evidently, on the whole of magical Britain. Magical misdemeanors from underage wizards go largely unpoliced in all magical households, explaining the fact that several familys, such as the Malfoys and the Weasleys have sent children to Hogwarts who openly acknowledge having learnt spells outside of Hogwarts. You may recall, once the ministry was in Voldemort's pocket, the trace was used to track down underaged wizards not in Hogwarts. I'd always supposed Hogwarts worked in conjunction, or possibly with special permissions, from the ministry to use the trace to send letters to children who had performed slight magical acts, such as Harry phasing his cousin through the glass at the snake exhibit.