If I use a MOGRT in a premiere file, can other editors see it if I share the project?

yes, if you are sharing the whole project by using the project manager option, then all the mogrt that were used will get collated by default to that folder. 

Assets typically need to be on each machine's storage. If you're all accessing them from the same shared storage such as a CC Library or your network server, then yes.

If not, no. Add them to a shared CC Library and apply them from there, and then ... yes