The Ministry probably doesn't monitor Hogwarts because students there have to do magic.
Monitoring every instance of magic used at Hogwarts would be a difficult task. The vast majority of the time, any magic detected would be an appropriate use of it, with it only occasionally being a student using magic in a way they shouldn't. In a normal school day, students would of course be doing magic, since the entire point of going is for the students to learn how to use their magic properly. In some classes, they start attempting spells in their first lesson, so it’s a necessary and frequent part of their learning.
“They were all very impressed and couldn’t wait to get started, but soon realised they weren’t going to be changing the furniture into animals for a long time. After making a lot of complicated notes, they were each given a match and started trying to turn it into a needle.”
- Harry Potter and the Philosopher's Stone, Chapter 8 (The Potions Master)
As far as monitoring Hogwarts for types of magic that are considered Dark, it seems unlikely that the Ministry would do that, since Hogwarts would have teachers and other staff who would be considered responsible for the behavior of the children, similarly to how the Ministry considers children in wizarding households under the care of their parents.
‘So if you’re under age and you do magic inside an adult witch or wizard’s house, the Ministry won’t know?’
“They will certainly be unable to tell who performed the magic,’ said Dumbledore, smiling slightly at the look of great indignation on Harry’s face. ‘They rely on witch and wizard parents to enforce their offspring’s obedience while within their walls.” - Harry Potter and the Half-Blood Prince, Chapter 17 (A Sluggish Memory)
Hogwarts seems to be given a fairly large degree of autonomy. The Ministry only really meddles with the happenings at Hogwarts in very serious situations (except when it was actually under control of the Dark Lord).