My son got a case of what they call 'mini-measles' after his MMR. It is quite rare but some kids get a mild case of the measles about two weeks after their shot. My son managed to get an "extreme" case according to the doctor and caused quite a bit of excitement for the local health department. He had the rash, fever and sensitivity to light.
As mentioned, these reactions are completely non-contagious as evidenced by the fact that the health department was concerned only with our son and not with anyone else that he came in contact with (including unvaccinated infants at playgroup).
Yes! My dd only got MMR and had a reaction. All over her face and neck. Hers showed before two weeks though. We think hers was an allergic reaction to the eggs since it showed so soon and wasn't the topical rash. He allergist and doctor both said if it was a rash from the shot, it would have been by the injection site not her face.