Mishnah 5 in Chapter 5 is structured in the same format as the previous Mishnah: we have two pieces of food, and three cases - A, B, and C. The Mishnah states: "Chatichah shel chullin v'chatichah shel chelev" - one piece is chullin, regular kosher food, and the second piece is chelev.
Eating chelev intentionally is punishable by kares, and therefore one who eats it inadvertently, without knowing that he is doing something forbidden, is obligated to bring a chatas. If he does not know whether he ate it or not, this is the classic case of doubt: there was a prohibition punishable by kares, and it is possible that he transgressed it and possible that he did not. Since he cannot bring a chatas, he brings an asham talui. This is the halachah.
The three cases in the Mishnah:
Reuven eats only one of the pieces.
Reuven eats both of them.
Reuven eats one, and Shimon eats the second.
"Achal achas meihen v'ein yadua eizo meihen achal" - Reuven ate one of the pieces and he does not know which of them he ate. In this case, "meivi asham talui" - he brings a regular, familiar asham talui, for perhaps he is obligated to bring a chatas and perhaps he is not, and since the matter is unknown - he brings an asham talui. Everyone agrees to this.
"Achal es hasheniyah" - if Reuven went and ate the second piece as well, then it is known with certainty that he ate chelev, and therefore the halachah is simple: "meivi chatas" - he brings a chatas, for he inadvertently transgressed the prohibition of eating chelev. In this too, everyone agrees.
"Achal echad es harishonah uva acher v'achal es hasheniyah" - Reuven ate one piece and Shimon ate the second. We do not know who ate what, but certainly one of them ate the chelev. The Mishnah says: "Zeh meivi asham talui v'zeh meivi asham talui, divrei Rabbi Akiva" - this one brings an asham talui and this one brings an asham talui, these are the words of Rabbi Akiva. Each of them is in doubt as to whether he transgressed the prohibition, and therefore brings an asham talui in the meantime. The Sages also agree to this.
Rabbi Shimon, following his opinion in the previous Mishnah, holds otherwise: even though each of them is in doubt, between the two of them one certainly ate the chelev, and one of them is certainly obligated to bring a chatas. Therefore, instead of each bringing an asham talui, the two of them should bring one chatas in partnership and on condition, and each will say: 'If I am the one who ate the chelev - it is mine, and if not - I declare my portion ownerless.' Therefore, "sheneihem meivi'in chatas achas" - both of them bring one chatas - one definite chatas between the two of them, a standard chatas.
Rabbi Yosi says, as he stated above: "Ein shenayim meivi'in chatas achas" - two people cannot bring one chatas. There is no reality of a partnership in a chatas, because a person is required to know the sin which he committed. One who does not know what he did cannot bring a chatas. Therefore, Rabbi Yosi holds like the Tanna Kamma, like Rabbi Akiva, and such is the halachah: each one brings an asham talui, for it is possible that he transgressed the prohibition of eating chelev and it is possible that he did not transgress.
In summary: In this Mishnah, three cases were discussed regarding two pieces, one of chullin and one of chelev: one who eats one of them and does not know which - brings an asham talui; one who eats both of them - brings a chatas; and if one ate the first and his friend came and ate the second - according to Rabbi Akiva, each brings an asham talui, according to Rabbi Shimon, both bring one chatas in partnership, and Rabbi Yosi disagrees and holds that two people cannot bring one chatas, and such is the halachah.