Must I share everything?

It doesn't matter if we've just stuffed the kids full, once we sit down with a plate of food...they want some! It's shocking to me that even after they have eaten until they are completely full, if they see us eating something (anything for that matter) they want some. We can be eating something that they wouldn't touch in a million years if we put it on their plates, and if it's on our plates, they say, "Mmmmm! Bite!" They are like little pigeons attacking! And when 3 of them come attacking, you better be willing to share!