Naturalization at Training Sites Act of 2017 or the NATS Act This bill directs the Department of Defense, and the Department of Homeland Security with respect to the Coast Guard, to establish a naturalization office with trained personnel at each initial military training site of the Armed Forces. Each military department concerned shall: (1) identify members of the Armed Forces who are not U.S. citizens and inform them of the availability of naturalization services at such offices, and (2) assign new non-citizen members of the Armed Forces to an initial military training site that has such an office. Naturalization notifications shall be completed: (1) during every stage of basic training, (2) during training for any military occupational specialty, (3) at each school of professional military education, (4) upon each duty station transfer, and (5) at any other time determined appropriate by the military department concerned. U.S. Citizenship and Immigration Services shall annually publish the number of members of the Armed Forces and their spouses and children who became naturalized U.S. citizens during the most recent year for which data is available, categorized by the country in which the naturalization ceremony took place.