The MMR vaccine is given subcutaneously (goes into the fatty layer beneath the skin) and as the following paper shows, children vaccinated with it, are able to shed the measles vaccine, as it can appear in the throat, and can be excreted via the respiratory system.

Morfin F, Beguin A, Lina B, Thouvenot D. Laboratory of Virology, Hospices Civils de Lyon, Domaine Rockefeller, 8 avenue Rockefeller, 69373 Lyon Cedex 08, France. Detection of measles vaccine in the throat of a vaccinated child. Vaccine. 2002 Feb 22;20(11-12):1541-3.

Measles vaccine is widely used, most often in association with mumps and rubella vaccines. We report here the case of a child presenting with fever 8 days after vaccination with a measles-mumps-rubella vaccine. Measles virus was isolated in a throat swab taken 4 days after fever onset. This virus was then further genetically characterised as a vaccine-type virus. Fever occurring subsequent to measles vaccination is related to the replication of the live attenuated vaccine virus. In the case presented here, the vaccine virus was isolated in the throat, showing that subcutaneous injection of an attenuated measles strain can result in respiratory excretion of this virus.