They Knew Newspapers
The original script for the play The Front Page was written by two newspaper guys, Ben Hecht and Charles MacArthur. MacArthur had worked at the City News Bureau of Chicago before he went on to work in screenwriting. Hecht worked for the Chicago Daily News as a war reporter—and as a crime reporter (Hildy is a crime reporter in The Front Page).
Hecht went on to be a famous screenwriter, novelist, and all around mover and shaker. (He's even supposed to have ghostwritten Marilyn Monroe's autobiography My Story, though he denied it). Richard Corliss, a film historian, said "Ben Hecht was the Hollywood screenwriter" and added "Hecht personifies Hollywood itself."
But though he personified Hollywood, and wrote snow drifts of film scripts, Hecht didn't write this film script. (He did chip in a little, though.)
He Knew Hollywood
Instead, His Girl Friday was adapted by Charles Lederer. Lederer had worked with Hecht on the 1931 film adaptation of the play The Front Page, so he had a running start on this second adaptation. (The Front Page got adapted over and over. It was like a superhero franchise, but with fewer tights and better hats.)
Lederer was a lifetime Hollywood dude. His parents were both in theater, and when they divorced he was raised in Hollywood by his aunt, actress Marion Davies. Film was in his blood; snappy dialogue was in his DNA.
And His Girl Friday is perhaps his most famous collaboration with Howard Hawks. (The other one that's in the running is Gentlemen Prefer Blondes, starring Jane Russell and Marilyn Monroe. Yup: Lederer was a bigwig.)
Not That Guy
Hecht and Lederer were famous, accomplished, and brilliant zingers flew out of their mouths whenever they opened them. But the most brilliant piece of writing on the film didn't come from either of them. It came from Howard Hawks.
The story is that in an early reading of the script, Hawks had a female assistant read the part of Hildy Johnson. He liked the way it sounded, and so decided to make Hildy a woman rather than a man. That change transformed His Girl Friday from a newspaper story to a romantic comedy about working women and the battle of the sexes. It's one reason why His Girl Friday is great, rather than just really good.
And the other reason why this script is so super-memorable? That's because of the way the lines are delivered: in super fast overlapping dialogue. That's a Hawks trademark.