Counterterrorism Screening and Assistance Act of 2019 This bill directs the Department of State to report biennially on U.S. initiatives to build capacity in foreign countries to prevent travel by terrorists and foreign fighters, and accelerates programs to share equipment with countries to prevent such travel. The report shall include (1) an assessment of the countries of greatest concern, (2) data such as the number of flights from such countries to the United States, (3) information on existing U.S. programs to build capacity to prevent travel by terrorists in those countries, and (4) a plan for prioritizing U.S. resources toward the countries of greatest concern. The Department of Homeland Security (DHS) shall accelerate provisions to foreign governments of (1) U.S. Customs and Border Protection's global travel targeting and analysis systems and (2) the State Department's watch-listing, identification, and screening systems. DHS may provide equipment, training, and maintenance services related to such systems. DHS shall notify Congress when it provides such equipment to a foreign government, along with information such as the specific risk that will be mitigated by the system and the total dollar value of the equipment provided. The bill directs the State Department to report to Congress each year on countries that fail to meet minimum standards to combat terrorist and foreign fighter travel. The State Department is authorized to suspend certain foreign assistance to countries that do not make significant efforts to meet such standards.