This משנה is dealing with a case where we know that all of the עדים required by the husband did in fact sign (cf. משנה ח, where it is assumed that a גט קרח is lacking the signatures of witnesses who were required by the husband).  In such a case, we need signatures merely for appearance's sake, that is to prevent another ב"ד from invalidating it for missing signatures.  Thus, according to בן ננס, one of these 'missing' signatures can be done by someone, including a slave or a thief, who would normally not be accepted as a witness.   ר"ע limits it to relatives who are otherwise כשר להעיד, as we do not want someone reading this גט to later think that a person who is a slave or thief is כשר להעיד.

רבי אליעזר מתיר - He bases his position on (ואשה גרושה מאישה לא יקחו (ויקרא כא, ז which means, regarding a כהן, that "and a woman divorced  from her husband he shall not take." The implication, according to ר"א, is that even if she is not permitted to every other man, she is still divorced.

וחכמים אוסרין - They hold that as it says that (וכתב לה ספר כריתת (דברים כד, א, this means that a divorce must be total, and that being forbidden to even one other man is not כריתת.  With regard to ואשה גרושה מאישה לא יקחו, that serves as a basis for the special אסור on כהנים known as ריח הגט.  Pursuant to that הלכה, even though she is not divorced by the גט of this משנה, she may no longer marry a כהן.