This is one of the most famous "plot holes" in cinema history, but the film actually provides a specific answer that many viewers miss during the first watch.
While the opening shot shows only the nurse entering the room after the word is spoken, the mystery of how it got out is addressed during the interview with Raymond the Butler:
The Eyewitness: Near the end of the film, Raymond explicitly tells the reporter, Jerry Thompson, "I heard him say it". He describes the scene exactly as the audience saw it: Kane saying the word, dropping the glass ball, and it breaking on the floor.
The Perspective Theory: Many film historians argue that the opening scene is actually shown from Raymond's point of view. Since Raymond was the one "taking care of him," he would have been in the room or just outside the door.
A Recurring Phrase: Raymond also mentions he heard Kane say "Rosebud" one other time—years earlier, right after Susan left him and he trashed her room. This established that the word was already a part of Kane's private vocabulary around the staff.
The Script vs. The Screen: In the original shooting script, the nurse was intended to be in the room when he said it. The "plot hole" only exists because Orson Welles chose a more stylized, eerie shot for the final film that made the room look empty for dramatic effect.
So, while it looks like he was alone, the movie's own internal logic says the butler was the one who leaked the word to the press.

I believe the butler also mentions he was in the room when he died.
So many stories, so little time.

Ok, thanks for clarifying!

Yes, as she came into his room.

Are you sure?

His nurse did