Jose Señarís Romay was at work at a production studio in Madrid in late March when he received an urgent text message. An attorney representing HBO was outside with a contract for him to sign. It said, in essence, stop divulging plot twists of the final season of “Game of Thrones.”

Filming for the HBO series—the show wraps up Sunday after eight seasons—has been shrouded in secrecy. Cast members say digital scripts disappear once a scene is shot. Actors who dub the show in other languages say they had to watch their characters...