I was covering BYU baseball for the Daily Universe and I would sit next to the great SID Ralph Zobell up in the press box. Ralph also acted as the game's scorekeeper. There was a ball hit that was kind of a tweener--could have been scored as a hit or an error. Ralph looked at his assistants and said, "What do you think, boys? Error?" and they nodded. They just decided it right then and there. I don't know why I never knew that's how it works, but I was kind of surprised. I guess I was expecting some kind of super computer to make the determination.
Years later, I read a story about Russell Westbrook working the scorekeepers over to count tips as rebounds during his tear of triple doubles in the NBA. There was a big story in The Ringer (or maybe Grantland) about how he would touch the ball off a missed shot and he'd scream "rebound!" at the scorekeepers. Kinda funny.
Anyway... scorekeepers are the real power brokers behind the stats we all love so much.