Gemara

(Mishnah): If a Get was written in Kesav Yado (the husband's handwriting), but witnesses did not sign it, or if only one witness signed the Get; it is Pasul. If she remarried, the children are Kosher (not Mamzerim)

R. Elazar says, even if no witnesses signed, but it was given in front of witnesses, it is valid. Witnesses sign a Get only due to an enactment for Tikun ha'Olam.

(Rav): The Mishnah discusses a Get (with one witness) in Kesav Yado.

Had a scribe written it, the children would be Mamzerim.

86b (Rav): The Halachah follows R. Elazar regarding Gitin.

R. Elazar holds for all documents that Edei Mesirah Karsei (the witnesses who see a document handed over empower it). Rav holds like him only for Gitin.

(Mishnah): If two men sent identical Gitin, and they got mixed up, each Shali'ach gives both Gitin to the wife he was sent to.

(R. Yirmeyah): This is unlike R. Elazar, who says that Edei Mesirah Karsei. Here, the Edei Mesirah do not know which Get divorces each woman!

Answer #2 (Abaye): The Mishnah can even be like R. Elazar. He requires only that the Get be written Lishmah, but it need not be given Lishmah.

Rishonim

Rif: The Halachah follows R. Elazar in Gitin. When Edei Chasimah signed a Get , we do not need Edei Mesirah. Rav rules like R. Elazar, yet he disqualifies the child of one who remarried based on a Get with one witness and Kesav Yad Sofer. If there were Edei Mesirah, the Get is Kosher without any Edei Chasimah! One signature would not disqualify the Get and the child! Rather, there were no Edei Mesirah, and even so, had two witnesses signed, the Get would be Kosher. Witnesses sign a Get only for Tikun ha'Olam. I.e. if there were only Edei Mesirah, perhaps they would go away and she could not prove that he divorced her. If Edei Chasimah do not suffice by themselves, there was no Tikun, for they do not prove that there were Edei Mesirah! L'Chatchilah we require Edei Mesirah. They are primary. A Gaon disqualifies one Ed Chasimah and Kesav Yad Sofer due to Mezuyaf Mitocho (the witnesses are invalid). This is unlike I wrote.

Rebuttal (Rosh 9:7): 1) Rav explained the (Reisha of the) Mishnah, which is like R. Meir. The Halachah follows R. Elazar. He does not expound "v'Chosav" to refer to the signing. Rather, he requires Edei Mesirah. Everywhere, he says Edei Mesirah Karsei. This implies that the divorce depends only on them. Without them, it is not a Get at all. 2) Witnesses sign only for Tikun ha'Olam, i.e. there should also be Edei Chasimah. When she holds a signed Get, we assume that it was given properly with Edei Mesirah. Everyone knows that Devar sheb'Ervah requires two witnesses. A Get permits her only if it was given in front of two witnesses. R. Tam explains that even R. Meir, who says that Edei Chasimah Karsei, requires that it is given in front of two, for Edei Chasimah do not testify that he gave it to her. 3) The Rif says that Edei Mesirah are primary. What does this mean? Mid'Oraisa, either suffice! Surely, he cannot give a Get when they were alone and permit her. 4) R. Yirmeyah thought that the Mishnah of Gitin that became mixed is like R. Meir. If all agree that Edei Chasimah suffice, why did we think that it is only like R. Meir? (The answers to these questions are according to the numbers.)

Defense (Milchamos Hash-m and Sefer ha'Zechus 47b): 1) If Rav explained the first Tana, the Gemara would have asked that Rav rules like R. Elazar! 2) We cannot assume that a signed Get was given in front of witnesses. Not everyone knows the Halachah, and not everyone divorces in Beis Din. If the husband will say that he gave it without witnesses, can we ignore him?! We are concerned if he claims that a Get from Chutz la'Aretz was Lo Lishmah, even though scribes write Gitin and they are proficient! (According to the Rosh), why do we need Edei Chasimah or Edei Mesirah? It should suffice that witnesses saw it in the husband's hand, so he cannot claim that she forged it! Surely, a Get in his handwriting should suffice! 4) R. Yirmeyah thought that just like R. Meir requires signing Lishmah, R. Elazar requires giving Lishmah, since Edei Mesirah Karsei. "Lah" refers not only to the writing, rather, also to the giving, which is Kores. We do not rule like R. Elazar regarding monetary documents. How can it be that Edei Chasimah do not help for Gitin, but work for money? In both cases it says "Sefer"! Why didn't we learn the argument about a signed Get given without Edei Mesirah?

Defense (Ran 47b DH v'Hainu and 48a DH ume'Atah): The Gemara said that if witnesses signed Lo Lishmah, R. Eliezer agrees that it is Pasul, for it is Mezuyaf Mitocho. If Edei Mesirah are required in any case, why is it Pasul? R. Eliezer always requires Edim on the Mesirah. However, when a signed Get is given, the Edei Chasimah become Edei Mesirah, for we know through them that he gave the Get to her. 3) Real Edei Mesirah, in front of whom the Get was truly given, are primary.

Rambam (Hilchos Gerushin 1:13): A Get must be given in front of witnesses, for it says "Al Pi Shnei Edim Oh Al Pi Sheloshah Edim Yakum Davar." It cannot be that today she is Ervah, and one who has Bi'ah with her is killed, and tomorrow she is permitted, without witnesses. Therefore, if he gave it when they were alone, or even with one witness, it is not a Get at all.

Rambam (15): Chachamim enacted that witnesses sign a Get. Perhaps he will give a Get to her in front of two, and they will die and her Get is like a shard, for it has no witnesses. Therefore they enacted that witnesses testify in the Get. Even though there are witnesses in it, he gives it to her in front of witnesses, be they the Edei Chasimah or different witnesses, for Edei Mesirah are primary.

Poskim

Shulchan Aruch (EH 133:1): There must be two Kosher witnesses when he gives the Get (Rema - and he gives the Get in front of them together). Similarly, he must sign witnesses on it.

Rebuttal (Gra 1): Tosfos (3b DH Rebbi) and the Rosh (4:12) say that Chachamim encourage to have Edei Chasimah, but it is not obligatory.

Shulchan Aruch (ibid.): B'Di'eved, if there were only Edei Chasimah and he gave it when they were alone, it is Kosher. The same applies if there were Edei Mesirah, but no Edei Chasimah. Some say that if we know that he gave it without Edei Mesirah, even if witnesses signed on it, it is Pasul. However, when we see it signed, we assume that it was given with Edei Mesirah.

Beis Yosef (DH veha'Rosh): Tosfos (4a DH d'Kaima), Rashi (64a), R. Efrayim, Ba'al ha'Ma'or, Semag, Sefer ha'Terumos and the Mordechai hold like the Rosh. The Rambam, Ra'avad, Ramban, Rashba, Ran and R. Yerucham hold like the Rif.

Beis Shmuel (3): If there were only Edei Chasimah, Tosfos, the Rosh and Mordechai say that it is Batel. Shiltei ha'Giborim disqualifies to Kehunah, lest it be confused with a Kosher Get. Some Ge'onim say that it is Pasul, but she need not leave her new husband. In practice, this requires investigation.