Gemara

20b (Mishnah): Regarding Terumah, if one hand became Tamei, the other is Tahor. For Kodesh he must immerse both;

This is because one hand is Metamei the other for Kodesh, but not for Terumah.

24a (Rav Shizbi): This refers to b'Chiburin (if they touched, or were connected; this will be explained).

Inference: When they are not b'Chiburin, one is not Metamei the other.

Question (Abaye - Beraisa - Rebbi): A dry hand is Metamei the other for Kodesh, but not for Terumah;

R. Yosi b'Rebbi Yehudah says, it is Posel, but it is not Metamei.

Summation of question: If one hand is Metamei the other even not b'Chiburin, it is a Chidush that this is even if they are dry. But if this is only b'Chiburin, what is the Chidush that it is even if they are dry?

(Reish Lakish): Reuven's hand is Metamei his other hand, but not Shimon's hand;

(R. Yochanan): The hand that itself touched Tum'ah can Metamei his other hand or another's hand;

It is Posel, but it is not Metamei.

Question: What is R. Yochanan's source?

Answer (Seifa): One hand is Metamei the other for Kodesh, but not for Terumah.

Question: The Reisha already taught this!

Answer: The Seifa teaches that one hand is Metamei another's hand for Kodesh.

Reish Lakish retracted and agreed with R. Yochanan.

Tana'im argue about whether or not a hand is Metamei another, or only Posel:

(Mishnah - R. Yehoshua): Anything that is Posel Terumah is Metamei hands to be Shniyos. One hand is Metamei the other;

Chachamim say, hands are Shniyos. A Sheni does not make a Sheni.

Inference: A Sheni does not make a Sheni, but it makes a Shelishi!

Rejection: Perhaps a Sheni does not make a Sheni or a Shelishi.

Rather, the following Tana'im argue about whether or not a hand is Metamei another, or only Posel:

(Beraisa - Rebbi): A dry hand is Metamei the other for Kodesh, but not for Terumah;

R. Yosi b'Rebbi Yehudah says, it is Posel, but it is not Metamei.

Yadayim (Mishnah 2:3): If one poured Mayim Rishonim on one hand and then poured Shniyim on both hands, both of his hands are Temei'im. If one washed one hand and rubbed it on the other, it is Tamei.

Rishonim

Rashi (DH b'Chiburin): One hand is Metamei the other b'Chiburin, i.e. the hands are touching when the Tahor hand touches Kodesh.

Tosfos (DH b'Chiburin): B'Chiburin means that the hands were touching when the Tum'ah touched one hand.

(Ba'al ha'Ma'or): B'Chiburin means that the Tamei hand touched the other. Lo b'Chiburin means that one is Metamei the other even without touching. This can happen in two ways:

Tum'ah touched one hand.

Both hands were Temei'im and one was immersed. It is a bigger Chidush that even in this case, the dry hand (the one that was not immersed) is automatically Metamei the other. The Beraisa refutes Rav Shizbi, for if one hand is Metamei the other only through touching, it makes no difference how this happened, so the Beraisa should not have mentioned a dry hand.

Rambam (Hilchos Mikva'os 11:5): If one poured Mayim Rishonim on one hand and then poured Shniyim on both hands, both of his hands are Temei'im. The Shniyim become Temei'im from the second hand (that was not washed before), and that water is Metamei the first hand. If one washed one hand and rubbed it on the other, the water on it becomes Tamei from the other hand, and it is Metamei the washed hand.

Hagahos Maimoniyos (Berachos 6:8): likewise, if Reuven did not wash and he touched Shimon's hands that were still wet from washing, Shimon must dry his hands and wash again, for Reuven was Metamei them through the water on them.

Rosh (Chulin 8:18): The Mishnah says that if one washed one hand and rubbed it on the other, it is Tamei. This is even if poured Shniyim on both hands together. This is because the water on the first hand became Tamei, and it was Metamei the second hand, and Mayim Shniyim. Therefore, if one poured water on one hand and rubbed it on the other, he must dry his hands (and wash again). If he poured a Revi'is on each hand, we are not concerned for this, for then Shniyim are not needed to Metaher the Rishonim.

Piskei Tosfos (Sotah 1:5): The custom is to pour water on one hand and rub it on the other (to clean them, before Mayim Rishonim). However, if he intended to wash only one hand and rubbed it on the other, it is Tamei.

Poskim

Shulchan Aruch (OC 162:4): If one poured water on one hand and rubbed it on the other, the washing did not help, even if he later poured water on both hands. This is because the water on the first hand became Tamei, and it was Metamei the second hand. Mayim Shniyim is Metaher only water that became Tamei from that hand itself. Therefore, if one poured water on one hand and rubbed it on the other, he must dry his hands and then wash properly.

Magen Avraham (9): This connotes that even if he poured a Revi'is on one hand and touched the other, the first becomes Tamei. But Sa'if 1 connotes that a Revi'is never becomes Tamei! We must say that water is prone to become Tamei at the time of Netilah in the place of Netilah (on the hand). Alternatively, since there is not a Revi'is on the second hand, it becomes Tamei.

Mishnah Berurah (32,35): This is when he poured part of the Revi'is on one hand. It is whether he touched the second hand in order that the Netilah will count also for it, or to clean it from sweat, or without any intent. He must dry his hands, because no quantity of water is Metaher Tamei water (other than Rishonim) on his hands.

Kaf ha'Chayim (38): If he will immerse his hands, there is no need to dry them.

Kaf ha'Chayim (23): One may rub his hands together after pouring Mayim Shniyim on both hands.

Kaf ha'Chayim (25): Some (Piskei Tosfos) say that if one intended to rub his hands together from the beginning, the Netilah was valid. Therefore, when he dries and washes again he should not bless. Some say that one blesses, for the Beis Yosef says not to rely on this. Some say that even Tosfos discusses only when the hand did not need to be dried, and Tosfos would agree here. I say that whenever one must wash again, his first Berachah is valid for the second washing unless he took his mind off his hands, interrupted by speaking or must wait a while to get more water.

Shulchan Aruch (ibid.): If Reuven did not wash and he touched Shimon's hands that were still wet from washing, Shimon must dry his hands and wash again, for Reuven was Metamei them through the water on them. Therefore, when he pours water the third time, i.e. the second time after Mayim Rishonim (which is only to remove dirt - Sa'if 2), he may not allow one hand to touch the other until he pours on both.

Kaf ha'Chayim (31): Even if Reuven is a newborn baby, Shimon's hands are Temei'im.

Mishnah Berurah (48): Some say that if the Tamei hand touched the hand that washed twice, he need not dry them. Since the hand was already Tahor, Mayim Shniyim suffice. Others require drying them, like if someone else touched them.

Shulchan Aruch (ibid.): If he poured a Revi'is on each hand, we are not concerned for all this.

Magen Avraham (11): If someone else touched his hands, they are Temei'im.

Mishnah Berurah (49) and Sha'ar ha'Tziyon (39): Some say that if he poured a Revi'is on each hand, we are not concerned even if someone else touched his hands. Therefore, if one remembered after blessing (that someone else touched them), he washes again without a Berachah. If one poured less than a Revi'is on his right hand and while there was still (Tamei) water on the hand he picked up the Keli, it is Metamei the handle. This will be Metamei his left hand (when he takes the Keli to pour Mayim Shniyim on the right hand)! Therefore, one should pour a Revi'is each time, or pour twice on the right hand before pouring on the left.

Rebuttal (Chazon Ish OC 24:23): We are not concerned for Tamei Kelim unless he touches where it is wet. In any case all our Kelim (nowadays) are Temei'im! However, we are concerned if his hands became wet from drops on the faucet before pouring Mayim Rishonim or Shniyim on them, for then even Mayim Shniyim will not help.

Kaf ha'Chayim (27): One should dry the handle before taking the Keli. If he cannot, he should dry each his hand before pouring on it, and pour all three times on the right hand before pouring on the left.

Shulchan Aruch (6): If one washed each hand by itself and then reconsidered and touched his hands together and poured Shniyim on both of them together, his hands are Temei'im. This is because they become Temei'im when he touched them together. The water on each hand is Metamei the other hand and the water on it. The Mayim Shniyim does not Metaher water that became Tamei from the other hand. Rather, he must dry his hands and then wash both together.

Source (Gra DH Notal): This is from the Tosefta (Yadayim 2:2).

Beis Yosef (DH v'Chosav): The Rambam brings these Halachos in Hilchos Mikva'os regarding Terumah, but not regarding Chulin. He holds that Mayim Shniyim applies only to Terumah, but not to Chulin. Still, he should have brought the laws of one who washed one hand with less than a Revi'is and then rubbed it on the other or reconsidered to wash them together, for they apply to even to Chulin! We cannot say that he is not concerned at all about Tum'ah after Mayim Rishonim, for in Hilchos Berachos (6:15) he requires lifting the hands to prevent the water from going past the wrist and returning! Rather, it seems that even though Mayim Shniyim is not needed, other laws do apply, i.e. washing one hand and then touching the other or water that went past the wrist and returned. In Hilchos Berachos he disqualifies when water went past the wrist and returned. This shows that all the concerns for Mayim Temei'im in Hilchos Mikva'os apply here also.

Rema: All the more so one must not touch a hand on which he poured twice to a hand on which he poured once.