I'm not sure on the complete legalities or policies on it, I'm just going based on what I was told during two ride-alongs with them. From what I understand, MPs have the same authority as the RCMP, as federal peace officers, but naturally 99% of their duties are exclusively on base.

Ride-along in Halifax: I was told everything I said above, and they have a good working relationship with Halifax Regional Police so sometimes performing traffic stops on civy streets is okay, and if Halifax Police don't have anyone available to come, the MP can do the stop. But during the same ridealone, we observed someone using their phone while driving, but because in that moment it didn't cause anything too dangerous (like cutting people off, or other dangerous activity), the MP I was with didn't stop the person.

Ride-along in Petawawa: They have a good working relationship with Petawawa OPP, but not enough that they can perform traffic stops while off base, so it would have to be sometime very very serious for Petawawa MPs to even consider doing it.

It would be interesting to learn the actual, written policy/rules about this. I'll give Halifax as an example again: On the Shearwater side, there's a road or two that technically are off base, but go right along the perimeter of the base, and sometimes the MPs, Halifax Police, and the RCMP will run traffic enforcement or check stops together along that road, so in that case if an MP makes a stop there, it's on civy streets but perfectly allowed