Mishnah 8 is the last in chapter 5, and the last in the series of cases before us. All the cases in this series are identical in structure, and the last case among them deals with two pieces: "chatichah shel chelev vachatichah shel chelev nosar" - a piece of chelev and a piece of chelev that is nosar.
The first piece - chelev alone, which carries the penalty of kares, and therefore one is liable to a chatas for it.
The second piece - chelev that is also nosar: besides the prohibition of chelev in it, an additional prohibition applies to it because it was left over from a korban after its time. This is an added prohibition, a second reason that this chelev is subject to kares, and therefore one who eats it inadvertently brings two chatas offerings, one for the chelev and one for the nosar.
Case 1 - He ate one of them and it is not known which of them he ate:
Reuven ate one of the two pieces and does not know whether he ate only chelev or chelev that is nosar, and therefore he does not know if he must bring one chatas or two chatas offerings. Since one chatas is definitely incumbent upon him, he brings it; and since he is in doubt regarding a second prohibition for which one is liable a chatas, he brings an asham talui with it. And this is the language of the Mishnah: "mevi chatas va'asham talui" - he brings a chatas and an asham talui.
Case 2 - He ate the first and afterward ate the second:
If Reuven went and also ate the second piece, it is clear to us that he ate two pieces of chelev, one of which was nosar. If this occurred in two separate lapses of awareness - where they informed him between one eating and the next about the prohibition of chelev, and he again forgot and ate inadvertently - "mevi shalosh chataos" - he brings three chatas offerings: one for the first piece of chelev, one for the second piece of chelev, and one for the nosar that he ate.
Case 3 - Reuven ate one and Shimon ate one:
Both of them definitely ate chelev, and it is not known which of them ate the nosar. Therefore, "zeh mevi chatas va'asham talui vezeh mevi chatas va'asham talui" - this one brings a chatas and an asham talui, and this one brings a chatas and an asham talui. Each one brings a chatas since he definitely ate chelev, and each one brings an asham talui, lest he is the one who ate the nosar and lest he is not.
Rabbi Shimon follows his reasoning: since it is known with certainty that one of the two committed the transgression, he says, "zeh chatas vezeh chatas, ushneihem mevi'in chatas achas" - this one brings a chatas and this one brings a chatas, and the two of them bring one chatas. Each one brings a chatas for the piece of chelev he ate, and instead of each bringing two chatas offerings, four in total, they bring three: a chatas for each one, and a third chatas jointly owned by both, on the condition that it will count for whoever ate the nosar.
Rabbi Yosi again disagrees: there is no joining together, and one cannot bring a sacrifice out of doubt on condition, but rather each one brings his own. However, here Rabbi Yosi elaborates in his wording and adds a new point: "kol chatas shehi ba'ah al chet ein shenayim mevi'in osah" - specifically a chatas that comes to atone for a sin, two people may not bring it in partnership out of doubt. From the implication of his words, it sounds as if there is a chatas that does not come for a sin, and that one can be brought in partnership in a case of doubt. And what is it?
A chatas that does not come for a sin - the korban of a woman after childbirth:
A woman who gave birth brings a chatas as one of her two sacrifices, even if she miscarried. Two women, where one of them had a child and the second did not, and the matter became confused and it is not known which of them gave birth - one of them is liable to a chatas and the second is exempt. Since one of them definitely gave birth, the two of them together buy one bird chatas and offer it in partnership, on condition.
Rabbi Yose says: This partnership works, since this sin offering does not come for a sin, and there is no prohibition carrying kares that caused her to bring it. Why, then, does a woman after childbirth bring a sin offering? A famous Gemara in Tractate Niddah (page 31) explains that she swore not to be intimate with her husband again. It is possible that this reason is more aggadic in nature than halachic; but whatever the reason may be - whether due to the pain, her share in Eve's sin, or a scriptural decree - the point is that she brings a bird as a sin offering, but not because of a kares prohibition she transgressed inadvertently.
Therefore, according to Rabbi Yose, the two women bring one bird between them and say: "Whoever gave birth to the child - let this be a sin offering for her."
In summary: In the three cases before us - someone who eats one of two pieces, someone who eats both of them in two separate lapses of awareness, and two people who each ate one piece - the Tannaim disagree about partnering to bring sacrifices conditionally. Rabbi Shimon allows them to combine and requires a third, shared sin offering, while Rabbi Yose rejects such a partnership for any sin offering that comes to atone for a sin. He permits this only for a sin offering that does not come for a sin, such as the bird sin offering of a woman after childbirth.
Halachah: The halachah does not follow even Rabbi Yose. We do not bring sin offerings and guilt offerings in partnership and conditionally, not even a bird sin offering.