Gemara

35b (Mishnah): One may not eat milk that a Nochri milked without a Yisrael watching, but one may benefit from it.

Question: What is the concern regarding milk?

Suggestion: Perhaps he switched it with milk of a Tamei animal.

Rejection: (This would be obvious.) Milk of a Tahor animal is white, and milk of a Tamei animal is Yarok! (Note - most Meforshim say that Stam Yarok is yellow. Some say that it is green.)

Suggestion: Perhaps he mixed in milk of a Tamei animal.

Rejection: We could test this by trying to curdle it! Milk of a Tahor animal curdles, but milk of a Tamei animal does not curdle.

Answer #1: Indeed, one who wants to make cheese may buy milk from a Nochri. If it curdles, it is Tahor. The Mishnah forbids putting it in food (or drinking it).

Question: We can test it by trying to curdle some of it!

Answer: In curdling (even pure Tahor milk), some (liquid) whey remains, so we will not know if some Tamei milk was mixed in.

Answer #2: Even one who wants to make cheese may not buy milk from a Nochri. Some whey remains between the cracks (after curdling). Perhaps this is from Tamei milk that was mixed in.

39b (Beraisa): A Yisrael may sit outside the herd of a Nochri milking his animals and buy the milk. He need not be concerned.

Question: If there are no Tamei animals in the herd, this is obvious! If there are Tamei animals in the herd, why (are we not concerned)?!

Answer: There are Tamei animals in the herd. When the Yisrael stands, he can see the Nochri, but not when he sits. One might have thought that since he cannot see the Nochri when he sits, we should be concerned lest milk of Tamei animals was mixed in. The Beraisa teaches that this is not so, because the Nochri fears lest the Yisrael stand and see him.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 3:15): If one eats milk that a Nochri milked without a Yisrael watching, he is lashed mid'Rabanan.

Rambam (17): If a Yisrael sat outside a Nochri's herd, and the Nochri brought to him milk from the herd, even though there is a Tamei animal in the herd, it is permitted, even if he did not see him milk. This is if the Yisrael can see the Nochri milk when the Yisrael stands, for the Nochri fears to milk from the Tamei, lest the Yisrael stand and see him.

Mordechai (826): The Gemara said that if there are no Tamei animals in the herd, surely it is permitted. Some learn from here that if there are no Tamei animals in the herd, it is permitted even if no Yisrael sees him. This is wrong. If so, the Gemara should have asked 'if there are no Tamei animals in the herd, why must a Yisrael sit near the herd and see him?' Rather, the Gemara means 'if there are no Tamei animals in the herd, surely it suffices for a Yisrael to sit near the herd!' Also R. Peretz requires a Yisrael sitting near the herd at the beginning of the milking even if there are no Tamei animals in the herd, lest the Nochri put Tamei milk in the Keli before the Yisrael comes.

Beis Yosef (YD 115 DH u'Mah): Semak says similarly.

Taz (2): The Mordechai mentioned sitting and seeing. It seems that sitting is not enough. What was the question 'if there are Tamei animals in the herd, why (is it permitted)?!' The Yisrael sees! We must say that the questioner knew that he cannot see when he sits, but he did not know that the Nochri fears. Therefore, he asks why sitting helps. However, the Tur explicitly wrote like the Shulchan Aruch here, that he cannot see him. To answer the Mordechai's question, they must say that when he sits outside, he sees that they do not bring bring something Tamei in at the time of milking. The Bach equates the Tur with the Mordechai. This is wrong.

Poskim

Shulchan Aruch (YD 115:1): Milk that a Nochri milked without a Yisrael watching is forbidden, lest he mixed in milk of a Tamei animal. If a Nochri was milking in his house, and a Yisrael was sitting outside, if he knows that there are no Tamei animals in the herd, it is permitted even if the Yisrael cannot see the Nochri when he milks. If there is a Tamei animal in the herd, and a Yisrael sits outside and the Nochri milks for the sake of the Yisrael, even if the Yisrael cannot see the Nochri when the Yisrael sits, if when he stands he can see the Nochri, it is permitted. The Nochri fears lest the Yisrael stand and see him. He knows that Tamei milk is forbidden to Yisrael.

Beis Yosef (DH Lo): Semak says that some people who lodge with a Nochri are lenient to drink milk that he milked for himself, but this is wrong. There is no distinction between one who milked for himself or for a Yisrael. If a Yisrael did not see from the beginning, it is forbidden.

Taz (1): There was a reason to think that if the Nochri milked for himself, surely he would not mix in Tamei milk, for it cannot curdle.

Shach (3): When the Yisrael does not see, it does not matter for whom he milks. When a Yisrael sits outside, if the Nochri milks for the Yisrael, it suffices if the Yisrael can see the Nochri when the Yisrael stands. If the Nochri milks for himself, the Yisrael must see when the Nochri milks, since there is a Tamei animal in the herd.

Taz (3): All the more so, it is permitted if the Yisrael goes in and out. This permits even for a Kusi who slaughters.

Shach (4): The Prishah says that it suffices even if the Yisrael goes in and out from sitting outside the herd.

Rema: L'Chatchilah, the Yisrael must be there from the beginning of the milking and see that there is nothing Tamei in the Keli into which he milks. The custom is to be stringent and not milk into a Keli that a Nochri usually milks into, lest some drops of milk of the Nochri remain.

Gra (4): The concern is not lest something Tamei be mixed in, for there is nothing Tamei in the herd! Also, the Nochri would not profit from this?! Rather, the concern is lest he was not careful to check that no drops of Tamei milk were left in the Keli.

Rema: B'Di'eved, one need not be concerned for all this.

Taz (4): This is like R. Peretz, and like Isur v'Heter, who added that the Yisrael must see that there was nothing Tamei in the Keli at the beginning. However, 'b'Di'eved, one need not be concerned for all this' connotes that this is even if there is a Tamei animal in the herd. This is wrong. Even when there is nothing Tamei in the herd, Isur v'Heter forbids, lest calamity result. A Yisrael must be there from the beginning of the milking, and see that there is nothing Tamei in the Keli. The Beis Yosef brought from Hagahos Ashri that R. Tam permits b'Di'eved if no Yisrael was there. This is only if we know that the Nochri milks for cheese, but not to eat the milk. Rather, the Rema discusses when there is nothing Tamei in the herd. He says that l'Chatchilah we are not lenient like the Tur to permit if the Yisrael sits without seeing. Rather, he must see. B'Di'eved it is permitted if he stood outside without seeing. If he was not outside, surely it is forbidden. The Rema should have made this clearer. L'Halachah, one should forbid even b'Di'eved if the Yisrael did not see from the beginning, even if there is nothing Tamei in the herd.

Shach (8): Isur v'Heter requires that the Yisrael saw the milking from the beginning and saw that there is noithing Tamei in the Keli. He is lenient b'Di'eved only about if the Nochri's Keli was used. The other Poskim agree. The Prishah says that going in and out is only to make the Nochri fear. This is wrong. It also removes concern lest something Tamei was mixed in, for the Nochri would need to leave the herd in order to do so, and the Yisrael sees that he did not.

Shach (7): The custom is to be stringent even if a Yisrael did not see anything Tamei in the Keli. We are concerned lest there were drops of Tamei milk and the Yisrael did not notice.

Gra (5): The Rema is lenient b'Di'eved if the Yisrael did not check the Keli at the beginning (unlike the Shach), or if the Nochri's Keli was used.