The Baraisa taught that the word 'you' also excluded separating Terumah for someone else.

Question: Is the main reason because it is not his or because it belongs to his friend?

Where would that make a difference? If he declared his pile ownerless and then re-acquired it (having separated Terumah when it was ownerless) - if it must be his, it is not Terumah; if it should not belong to his friend, it is Terumah.

Suggestion: (A later Mishnah taught) If a person stole Terumah of Hekdesh and ate it, he must pay two fifths (one for Terumah, one for Hekdesh) and the principal, but a thief does not pay double payment to Hekdesh.

Who separated the Terumah? Wasn't it the Temple treasurer who was separating from something that didn't belong to him! (This shows that the Torah is only particular that it doesn't belong to another person.)

Rebuttal: No. What was the case of Terumah of Hekdesh? A Kohen consecrated his Terumah.

Question: R. Hoshiya taught in a Baraisa - '...both a Yisrael who consecrates his Tevel (untithed produce) and a Kohen who consecrates his Terumah' - since it's become Hekdesh, it's the Temple treasurer who will be separating the Terumah from the Tevel...?

Answer (R. Idi): That is because the treasurer is considered to be the owner.

This is unlike R. Yosi who said that the treasurer is considered to be another person.

(R. Zeira/ R. Acha/ R. Yosa/ R. Yochanan citing R. Yannai): Instead of saying "you'', the pasuk says "also you'' - this includes your agent. Just as you are Jewish, so too your agent must be Jewish. You are able to appoint an agent (to separate Terumah), but a gentile cannot appoint an agent. (The Gemara now clarifies the meaning of this.)

R. Yosi thought that it means that a gentile cannot appoint another gentile as an agent, but he could appoint a Jew.

Rebuttal (R. Zeira): The pasuk itself disproves that - 'you' can appoint an agent - isn't it referring to a Jew becoming the agent? Similarly, when the pasuk teaches that a gentile cannot appoint an agent, it even refers to him appointing a Jew.

Rejection (R. Hoshiya): A Baraisa supports R. Yosi (that a gentile can appoint a Jew as his agent) - 'Rabban Shimon ben Gamliel said - (If a Jew received a field from a gentile for sharecropping.) If this gentile would wish not to separate Terumah from his produce, he may do so' - (so the Jew would need consent from the gentile before he separates), This implies that if it was known that the gentile wishes to separate, the Jew could do it for him!

Rebuttal (R. Abba): (Really, a gentile cannot appoint a Jew as an agent.) Here, the Baraisa is referring to when the gentile guides the Jew's hands.