But to say "We are normal" in response to raydon's question "What is a normal person?", although grammatically correct, is not definitive enough to exclude everybody except raydon and me. But your proposed answer "We are both normal" is grammatically correct and the word 'both' does indicate two people to the exclusion of all others and limits the response to the questioner and me.
However, my answer to raydon's question was basically incorrect in so far as he asked "What is a normal Person?", thus posing the question "What characteristics define a normal person?", whereas, in true politician style, I answered the question "Which of the people here are normal?"
I am having serious second thoughts about my normality, as surely it can't be normal to carry on a conversation normality, I vs. Me and definitive definitions, can it? Especially at 11:45 at night.