OPINIONS: On every Yom Tov, every adult Jewish male is obligated to bring three types of Korbanos to the Beis ha'Mikdash: the Olas Re'iyah, the Shalmei Chagigah, and the Shalmei Simchah. The former two are brought once during the Yom Tov, while the Shalmei Simchah are eaten every day of the Yom Tov in order to fulfill the Mitzvah of Simchas Yom Tov, experiencing the joy of eating the meat of Korbanos throughout the festival.
Does the obligation of offering Shalmei Simchah require that one sacrifice a Korban Simchah and eat it, or does it require only that one eat from the meat of a Shalmei Simchah, even if he did not sacrifice it? If the Mitzvah is only to eat from the Korban but not necessarily to sacrifice it, it is reasonable to assume that one may fulfill his obligation by eating the meat of someone else's Korban Simchah. If, on the other hand, the Mitzvah also requires that one sacrifice a Korban specifically for the Shalmei Simchah, one must offer his own animal and cannot fulfill his obligation of Simchah by eating the meat of someone else's. (ARUCH LA'NER to Sukah 48a, DH b'Masnisin)
(a) The Gemara in Pesachim (71a) explains that the obligation of Shalmei Simchah applies even to the last night of Sukos, the night of Shemini Atzeres (but not to the day of Shemini Atzeres, according to Rashi; see, however, Rashi to Sukah 48a, DH l'Rabos). The DEVAR SHMUEL there cites commentators who prove from here that the Mitzvah merely requires that one eat the Korban but not that he sacrifice it. Since a Korban cannot be offered at night, and yet the Gemara says that the obligation of Simchah applies at night, it must be that the obligation of Simchah requires only that one eat the meat of the Korban.
TOSFOS in Pesachim (96b, end of DH Ta'un) indeed writes that a person fulfills his obligation of Simchah by eating his friend's Korban.
(b) However, the Gemara here is bothered by the question of how a person can fulfill his obligation of Shalmei Simchah with an animal which is not Chulin. The Halachah is that any obligatory Korban, such as the Shalmei Simchah, must come from Chulin and not from an animal which is already consecrated as a Korban (such as Ma'aser Behemah or an animal purchased with money of Ma'aser Sheni). The Gemara cites a verse which teaches that the Shalmei Simchah is an exception to this rule and may be brought from Chulin.
If the Mitzvah is to eat the Shalmei Simchah, then there is no obligation per se to sacrifice an animal as a Korban Shalmei Simchah, and, consequently, the animal should not need to come from Chulin! If the Mitzvah requires only that meat be eaten to provide Simchah, then the meat obviously may come from any Korban. Only when the Mitzvah requires that an animal be sacrificed must the animal be brought from Chulin and sanctified for its designated purpose. The Gemara here implies that the Mitzvah requires that one sacrifice a Korban for Shalmei Simchah, and thus one may not satisfy his obligation of Shalmei Simchah by eating the meat of another's Korban. (It is unlikely that the verse teaches that one is not required to sacrifice, but only to eat, a Korban for Simchah. Rather, the verse teaches that even Nedarim and Nedavos may be eaten as the Korban Simchah (as an exception to the normal rule), and not that the Mitzvah of Simchah is merely to eat, but not to offer, a Korban. This is the implication of Rashi to Pesachim 70a, DH Yotzei. See, however, Rashi here on 7b, DH Af Min ha'Ma'aser.)
This is also evident from the ruling of Rebbi Elazar in Pesachim (70b) that one fulfills the Mitzvah of Simchah only when he eats a Korban slaughtered on the festival. If the Mitzvah involves only eating meat of a Korban, what difference does it make if the Korban was slaughtered on Yom Tov or before Yom Tov? It must be that the Mitzvah involves offering the Shalmei Simchah as well as eating it.
However, if the Mitzvah of Simchah requires that one offer the Shalmei Simchah, why does the Gemara in Pesachim say that the Mitzvah of Simchah applies even at night? The Mitzvah requires that one offer a Korban, but a Korban cannot be offered at night.
It must be that the Mitzvah of Shalmei Simchah is to eat an animal which one has sacrificed as a Korban (and, according to Rebbi Elazar, the animal must have been sacrificed on the festival). (Apparently, the Simchah must come from a festival-related Korban.) In this sense, offering a Korban is obligatory. However, it is not obligatory to bring a Korban on every day of the festival, and certainly not on the last night of the festival. In fact, it may not even be obligatory to bring a Korban on the festival altogether. The Gemara in Pesachim (70a, 71a) cites opinions that one fulfills the Mitzvah of Simchah even by eating a Korban slaughtered on Erev Pesach (in contrast to the opinion of Rebbi Elazar cited above); as long as the animal is his Korban, it makes no difference whether it was offered on the festival itself or before the festival. The Mitzvah requires only that each person eat from his Korban every day (and night) of the festival.
Tosfos also seems to be in doubt whether one fulfills the Mitzvah of Simchah with another's Korban. In Sukah (47a, DH Linah), Tosfos asks the same question he asks in Pesachim (how can a person avoid bringing a Korban of some sort every day of the festival), but he does not suggest the simple answer which he proposes in Pesachim, that one may eat from another's Korban Simchah.
(See also the words of RAV SHACH zt'l in AVI EZRI, Hilchos Chagigah 2:3, who proposes that there is also a special Mitzvah to offer a Korban Simchah one time during the festival, in addition to the Mitzvah to eat meat from a Korban every day of the festival. The Aruch la'Ner (ibid.) makes a similar proposal. See also SHIMUSHAH SHEL TORAH (p. 277-279), who recounts the CHAZON ISH's response to this proposal, and Rav Shach's reply to the Chazon Ish. See also DEVAR SHMUEL to Pesachim 109a.)