What do you call somebody who asks a question and somebody who answers a question?
Questioner/Asker, Answerer, Answerer
Noun
answerer (plural answerers)
1. a person or thing that answers or responds
Also see here: Should I prefer "asker" or "questioner" for a person who asked a question?
May I suggest: Questioner/Respondent
You could consider Inquirer, Questioner or, for instance Interrogator. Meanings differ slightly, I'd normally go for one of the first two. The third can be used when someone asks multiple questions, i.e., interrogates someone else.
Edit: respondent has been suggested by others.