Yes, they
do monitor your usage.
No, they probably don't red flag a chart/record when a med student is admitted

Students (and residents) are not VIPs.
I've no doubt that they monitor my usage extensively. Which is to say, it gets translated into a bunch of 1s and 0s and put onto a database somewhere for later review should there be an issue.
I disagree with your second point, though. We are all entitled to privacy of our health information. Those most likely to be accessed without permission are those that are
interesting to a hospital's employees: attendings, staff, celebrities, nurses, and yes, students and residents. (The average hospital donor or potential donor, which is the more common usage of "VIP" in my experience, may not represent much extra risk if he or she is not a household name.)
Student patients may be at greater risk of having their information violated due to the nature of medical school classes: compared to others with unrestricted EMR access, they are younger, less mature, and have not been around long enough to see punitive action taken against those violating HIPPA. In my n=1 experience, violation of student protected information has occurred on at least 2 occasions. It's clearly a problem and if anyone is deserving of extra monitoring, students ought not be excluded because of their perceived lesser importance to your team. To suggest that student/resident patients deserve less protection than other employees is kind of offensive.
EDIT: Given that "violations" of various degrees of maliciousness are occurring continually, I'll grant you that hospitals are probably doing constant PR damage control. As such, celebrities and athletes and persons of public interest, whose information is most likely to be widely disseminated, probably merit extra protection from a practical standpoint. I still don't think it's necessarily "right" though.
Oh, they do random screens. They supposedly do them at my facility, but everyone who works here gets treated here as well, so it must be a complicated set-up. I've cared for, operated on, and accessed the records of physicians, nurses, residents, techs, managers, IT personnel, etc. Plus, I've done research with chart reviews, and I've seen even more records (not just a few of which I recognized).
I've performed hundreds, if not thousands, of mis-clicks, MRN mistakes, same-name mistakes, and so on. I have done (approved) chart reviews requiring access to >700 individual's charts (some of whom were attending physicians, some VIP including politicians and celebrities). If my activity's not been "flagged" as yet, I cannot imagine what I could possibly do to bring that about.
I do not know any more than the average person, but like a few other posters have suggested, I believe the only way to monitor this would be at the individual chart level. If they were to investigate the entirety of my activity and ask me to justify each access, I'd just pack my bags and show myself out. Luckily, the OP's hospital administration seems to be alone in their heavy handedness and inability to think critically about the problem.