Secure and Trusted Communications Networks Act of 2019 This bill establishes (1) a mechanism to prevent communications equipment or services that pose a national security risk from entering U.S. networks, and (2) a program to remove any such equipment or services currently used in U.S. networks. Specifically, the bill prohibits the use of federal funds to obtain communications equipment or services from a company that poses a national security risk to U.S. communications networks (e.g., Huawei Technologies Co. Limited, Zhongxing Telecommunications Equipment Corporation, or any subsidiary or affiliate of either entity). The Federal Communications Commission must publish and maintain a list of such equipment or services. In addition, the bill establishes the Secure and Trusted Communications Networks Reimbursement Program to supply small communications providers (i.e., providers with 2 million or fewer customers) with funds to offset the cost of removing prohibited equipment or services from their networks and replacing it with more secure communications equipment or services.