Hullin, Chapter Ten, Mishnah One
Deuteronomy 18:3 states, This then shall be the priests due from the people: Everyone who offers a sacrifice, whether an ox or a sheep, must give the shoulder, the cheeks and the stomach to the priest.
Our chapter is concerned with these gifts given to the priest.
1) The [law of] the shoulder and the cheeks and the stomach is in force both within the Land and outside it, both during the existence of the Temple and after it, in respect of unconsecrated animals but not consecrated animals.
2) For it might have been argued thus: if unconsecrated animals, which are not subject to the law of the breast and the thigh, are subject to these dues, how much more are consecrated animals, with are subject to the law of the breast and the thigh, subject also to these dues!
3) Scripture states, And I have given them to Aaron the priest and his sons as a due for ever (Leviticus 7:34) only what is mentioned in this passage shall be his.
Section one: The law that a person must give the priest the shoulder, cheeks and stomach of an animal that he slaughters is in force in all places and in all times. It applies only to unconsecrated (hullin) animals and not to those that have been consecrated to the Temple.
Section two: The mishnah now explains why one might have thought that it should apply to consecrated animals. From shelamim sacrifices, whose meat goes to the owners, the priest receives the breast and the thigh. Thus one might have arguedif the priest gets the shoulder, cheeks and stomach from unconsecrated animals but not the breast and thigh, all the more so he should get the shoulder, cheeks and stomach from an animal from which he does receive the breast and thigh.
Section three: To counter this argument, the Torah hints (midrashically) that from sacrifices, Aaron and his priestly descendents receive only the breast and thigh, and not the shoulder, cheeks and stomach.
Along with this midrash, it seems to me that the different priestly gifts serve to distinguish between sacrifices and non-sacrifices. The fact that the priests receive different gifts from different animals might help in distinguishing whether the gifts they are receiving must be treated as holy, or can be considered hullin (non-sacred).