The Screenwriter is responsible for the story. That means the structure, how the whole thing hangs together, the development of characters, the event choices. The Screenwriter is responsible for thematic material, which is what are we actually trying to say.
That can be something simple like there’s no place like home in The Wizard of Oz or it can be something complicated and dark like the rich get away with murder, which is the theme of Chinatown. It can also be something unbelievably nuanced like Annie Hall where Woody Allen is writing about a woman who’s a mess and then she meets a man who turns her into someone strong enough to leave him. That’s pretty incredible. That’s all writing.
Most people think Screenwriters are responsible for only dialogue. I would say that’s about five percent of it.