we say "chez moi" as well, but here it is better with the possessive pronouns and home, as it is about sharing something with the other, do you see what I mean?
Then also "chez moiest chez toi" is not harmonious, I'm not a pure linguist so I wouldn't know the name from this lack of vocal esthetism that makes the difference in the choise of the words in a sentence, although I can honestly assume the version I edited with is prefered in fluent French