As far as my understanding goes, As long as Mr. Potato Head does not have all pieces missing from the central object, he can still animate. This is evident in the first movie where RC slammed into him and all his pieces fell out, he stopped moving. Even without being attached to the central object, however, his mouth can still move and project voices. This is likely why MPH did not just break off all his pieces and stick them on the tortilla shell, rather, he did it one part at a time.

Most likely each piece operates under the potato's orders. In the scene with the tortilla, the brain in the potato was ordering the pieces to move independently. But he still needed his potato body, which was why they went back for it later on.