Who gives the NATOPS brief, IP or student, is one of those things that goes back and forth every few years and varies from squadron to squadron. If you're in a crew aircraft in your fleet squadron then sometimes the aircraft commander will give the brief (this is the most proper way) but sometimes delegate it to one of the crew, for some reason or another.
In the training command, if the student gives it then it takes longer, there are little mistakes to correct, and sometimes the individual squadron might have stuffed a few too many things into the brief (making it not brief at all...). If the IP gives it then that should mean fewer mistakes and faster, but then sometimes the student doesn't learn the details as well. One of my primary IPs gave most of the brief, but at different parts he had me pick it up (for example, "engine failure we will ___" gestures to me to recite immediate action items, then he picks it back up again). You wouldn't see that much or at all in a fleet squadron, but in the training command it can be a useful tool to keep a good pace and keep both people's brains engaged in the brief. In the fleet, the entire brief can be divided up between more than one person is it pertains to each role. If your platform has a mission commander in the back and a pilot in command up front then that's one example, SAR crewmen is another.
Sometimes you're in primary squadron brand X doing things one way and your friends are in brand Y are doing things a slightly different way. That's kinda how a lot of things go in a lot of places in aviation.
