My hospital does not offer extra pay when students are around. But it makes sense that they don't because clinical students are taught by their instructors, not the staff nurses. The students each have a patient or two that they learn off of, and ahead of time the instructor informs the staff nurse assigned to those patients what tasks the student will be doing (passing meds, baths, vitals, etc), but it does not fall on the nurse to do any teaching if they don't want to. Most nurses day shift nurses that I know like having students on the floor because they are extra hands, especially when it comes to ADLs.
Actual precepting - one on one agreed upon ahead of time - does include a $1/hr diff.
So if you're just talking about clinical students on the floor with an instructor, assigned to a patient, then, no, it is NOT the nurse's job to make sure the students are being taught. They have enough to deal with without having a constant shadow slowing them down. I'm on nights, and I'd be pissed if a student was thrust upon me. At the same time, I would love to grab an eager student (who didn't hound me) and show them a few things they may not get to see (IV starts, foleys, IV piggybacks, etc).