Maybe I misunderstand it, but you want to ask about why did your neighbour said; "Sizin tabak" instead of; "Sizin tabağınız"?
Actually you're right, if your neighbour says "Sizin tabağınız" is going to be more correct. Actually "Sizin tabak" sounds unnatural for a Turkish ear.
(Sizin)* tabak(plate)-ınız. / (Sizin)* tabağınız. (Your plate)
(Bizim)* mutfak(kitchen)-ımız / (Bizim)* mutfağımız. (Our kitchen)
Saying (Sizin/Bizim) is optional, because probably you know that the supplement also means 'Yours/ours'
hope it was helpful and I understand it right))