The gemara concludes that we pasken like R. Elazar that the eidim who witness the giving over the gett are considered the ones who make the divorce go into effect. There is a big machlokes rishonim (rif and ran at the end of the masechta - ran has a very mechudash approach that eidei chasima work as if they were eidei messira) whether R. Elazar agrees that the witnesses signed in the gett can effect the divorce or only eidei messirah. Tosafos here is of the opinion that only the witnesses who see the gett being given make the divorce effective. The difficulty with Tosafos approach is the statement of R. Elazar that "eidim sign on the gett for tikkun ha'olam" just in case the eidei messira die or are overseas, which implies that when eidei messirah aren't present, the eidei chasima work by themselves? Tosafos explains that really the eidei chasima don't do anything, but their presence is an indication that the gett was delivered properly in the presence of eidei messirah.
Based on Tosafos assumption that eidei chasima really don't do anything, Tosafos (d.h. modeh) asked why should their be a problem of mezuyaf mitocho when the gett was signed not lish'ma. Normally mezuyaf mitocho is a concern that you will rely on the passul eidei chasima elsewhere, but here when it is signed not lishma, there would be no problem relying on them elsewhere. To answer this question Tosafos was forced to say that we make a gezeira that if the signing will be done not lish'ma, one will also come to write not lish'ma. However, according to the approach that R. Elazar agrees to rely on eidei chasima also, the question doesn't begin. Since we sometimes need to rely on eidei chasima, even R. Elazar maintains that it must be signed lishma.
Tosafos suggests that even according to R. Meir who holds that eidei chasima make the divorce effective, will agree that it must be delivered in the presence of witnesses to accommodate for "ein davar sh'berva pa'chos mi'shnayim". Being that the source of 2 witnesses by ervah is learned from money, we should require 2 witnesses to be present by a kinyan as well, and not rely on the signatures in the contract alone. However, Tosafos makes a distinction between money where the signed contract can work based on the person admitting to the transaction, but this concept of "hoda'as ba'al din" won't apply by get because it is detrimental to her by forbidding her to the kohein. However, see Rashi on the mishna 86a (bach writes that it is not really rashi, just a hagaha) who seems to say that when a husband writes the gett, it is effective without any witnesses according to R. Meir, based on the concept of hoda'as ba'al din. Rashi seems to hold that this concept applies to gittin, and we don't consider it a detriment by assuring her to a kohein, because in her present state of a married woman she is anyway assur to the kohein (perhaps if the husband is a kohein then even rashi would agree).