The training center (part 142) or air carrier (part 121 or 135) selects who they want as instructors or TCE/check airman. They submit a request to the FAA to have the person approved for the position. If the FAA is happy with the choice they will issue a letter stating what the pilot can do. It looks good to have a CFI but I know people who are TCEs but have never held a CFI. They usually have a long history of training, either military or airline.