Yes.

I would just do the outside

A kick plate is to be installed on the "push to open" side of a door. In commercial applications a mop plate (4" to 6" tall) is sometimes installed on the "pull to open" side.

Put them on whichever side you're trying to protect. Could be both sides of a two-way swinging door, perhaps only on the push-to-open side of a one way swinging door.

I'm going to put a plate on both sides. :)

Thanks for the advice, folks.