Medical Device Safety Monitoring Act This bill amends the Federal Food, Drug, and Cosmetic Act to require the Food and Drug Administration (FDA) to support pilot projects in order to provide timely and reliable information on the safety and effectiveness of marketed medical devices. The projects must: (1) be designed to generate safety and active surveillance data, (2) inform support for safety and active surveillance activities, (3) be coordinated with a system for evaluating medical device technology that operates under a board with representation from consumer groups and device manufacturers, and (4) use electronic health data. The FDA may determine that a manufacturer's participation in a pilot project satisfies requirements regarding reporting or postmarket surveillance if the project captures adverse event information and the FDA has established procedures to publish safety information from the project. Not later than January 31, 2021, the FDA must evaluate real world evidence pilot projects, such as the ones supported by this bill, for their ability to inform decision-making and efficiently generate evidence about the safety or effectiveness of medical devices.