I've gotten in trouble before while exploring and then being accused of putting my nose where it shouldn't be. It can look like you're wasting your time looking for distractions, or trying to dig up dirt. Not everyone in all parts of your company is going to like you poking around. You also risk picking up new responsibilities if people catch wind that you might be offering help. This can be the start to a promotion and also a good way to just make your days longer for no additional compensation. Not all managers will respond to the leverage of cross department work you took on yourself, though they probably should.
That said, if you do it well, which is an art of it's own. I agree, the potential benefits outweigh the risks.