Gemara

Ahavah brei d'R. Zeira - Beraisa: In all Berachos, one who was already Yotzei can be Motzi, except for Berachos on bread and wine (or other foods). In these, if he was already Yotzei he cannot be Motzi.

Question (Rava): What is the law regarding ha'Motzi of Matzah (on Seder night) and wine of Kidush (on Shabbos night)?

Since these are obligatory, he can be Motzi;

Or, since the Berachah (on the bread or wine) is not solely for the obligation, he is not Motzi!

Answer: Rav Ashi said that Rav Papi would make Kidush for us, and when his sharecropper came, Rav Papi would make Kidush for him.

Beraisa: One may not bless over bread for guests unless he eats with them. One may bless for his household to train them in Mitzvos. Regarding Hallel and Megilah, even one who was Yotzei can be Motzi.

34b: If an individual did not bless (say Shemoneh Esre), another may not bless for him.

Beraisa - R. Gamliel: If the Shali'ach Tzibur is not Motzi the Rabim, why does he say Shemoneh Esre out loud?

Chachamim: He is Motzi one who is not Baki (does not know the Berachos).

R. Gamliel: Just like he is Motzi one who is not Baki, he is Motzi a Baki.

R. Yochanan: The Halachah follows R. Gamliel regarding the Berachos of Rosh Hashanah and Yom Kipur.

Rishonim

Rif and Rosh (3:13): In all Berachos, one who was already Yotzei can be Motzi, except for Berachos on bread and wine. In these, if he was already Yotzei he cannot be Motzi, except for ha'Motzi of Matzah and wine of Kidush ha'Yom. One may not bless over bread for guests unless he eats with them. One may bless for his household to train them in Mitzvos. Regarding Hallel and Megilah, even one who was Yotzei can be Motzi.

Rosh (ibid.): All Berachos of Hana'ah are like the Berachos on bread and wine. If one makes a Berachah of Hana'ah for others, this is l'Vatalah.

Ran (DH Tani): One can be Motzi another because all of Yisrael are Arevim (responsible) for each other. As long as Reuven was not Yotzei, it is as if Shimon was not Yotzei. This does not apply to Berachos of Hana'ah, for they are not obligatory; one need not benefit! There was no question about the Berachos Al Achilas Matzah and Mekadesh ha'Shabbos. These are obligatory, surely he can be Motzi. The question is about ha'Motzi and Borei Peri ha'Gafen. These Berachos are not because of the Mitzvah, rather, because the person benefits.

Ran (11b DH Yachid) and Hagahos Ashri: The Yerushalmi gives three exceptions in which one was Yotzei cannot be Motzi - Birkas ha'Mazon, Kri'as Shma and Shemoneh Esre. In these he can be Motzi only if he himself is Yotzei now. Since it says "V'Ochalta v'Sovata u'Verachta", the one who ate should bless. In Kri'as Shma we say "V'Shinantam"; it is fitting for everyone to say the words himself. It is fitting for everyone to request mercy for himself, in Shemoneh Esre. However, if one is not Baki, the Gemara (Berachos 45b) says that a Baki is Motzi one who is not Baki.

Rif and Rosh (Pesachim 27b and 10:36): Normally, if he was already Yotzei the Berachah on bread or wine he cannot be Motzi others. There was a question if this includes ha'Motzi of Matzah and wine of Kidush ha'Yom. We learn that they are not included from Rav Papi, who would make Kidush again when his sharecropper came. The same applies to Borei Peri ha'Adamah of Maror, for it is also a Mitzvah.

Rambam (Hilchos Berachos 1:10): Regarding all Berachos, even if one already blessed and was Yotzei, he can be Motzi others. The exceptions are Berachos of Hana'ah that are not Mitzvos. Then, he can bless for others only if he benefits with them. In Berachos of Hana'ah that are Mitzvos, such as ha'Motzi of Matzah on Pesach night and wine of Kidush ha'Yom, he can bless for others and they will eat and drink, even though he does not eat with them.

Poskim

Shulchan Aruch (OC 167:19): One who is not eating may not bless ha'Motzi for those who eat. One may bless for children to train them in Mitzvos.

Beis Yosef (DH v'Ika): The Beraisa says that one may bless for his own household, but not for others' children, since their parents train them in Mitzvos. Alternatively, one may bless even for others' children. The Beraisa discusses the usual case.

Magen Avraham (40): The same applies to other Berachos of Hana'ah. Arvus does not apply to because they are not obligatory.

Mishnah Berurah (91): This is even if the others do not know how to bless.

Shulchan Aruch (213:2): One cannot bless (on food) for others unless he eats with them. Then, they are Yotzei by hearing even if they do not answer Amen.

Magen Avraham (7): If he does not eat with them, his Berachah is l'Vatalah, so surely they are not Yotzei. The Keneses ha'Gedolah disagrees.

Mishnah Berurah (15): If he did not eat with them due to Shogeg or Ones (e.g. his wine spilled), they are Yotzei.

Mishnah Berurah (17): Even if one did not answer Amen, Shome'a k'Oneh and it is as if he blessed himself. One must answer Amen to any Berachah he hears from a Yisrael. There is a bigger obligation when he wants to be Yotzei to answer, to actively show that he agrees with the Berachah.

Shulchan Aruch (273:4): One may make Kidush for others even if he does not eat with them, since for them it is b'Makom Seudah. Even though he cannot be Motzi them in Borei Peri ha'Gafen unless he eats with them, here it is obligatory like Kidush, therefore he can Motzi them.

Rema: This is permitted even in the morning Kidush.

Beis Yosef (DH u'Mah she'Chosav d'Af): The Mordechai says that he cannot be Motzi them in the morning Kidush, since it is only the Berachah Borei Peri ha'Gafen. R. Yerucham permits, since it is a Mitzvah. We follow the Rif and Rosh, who permit regarding Karpas.

Taz (3): Surely, if one makes Kidush at night for others on bread and does not eat with them, he can say ha'Motzi for them, for it is in place of Borei Peri ha'Gafen. Otherwise, he cannot say ha'Motzi for them. The Levush permits ha'Motzi even when Kidush was on wine, for Kidush must be b'Makom Seudah, so ha'Motzi is obligatory. This is difficult, for all three meals of Shabbos are obligatory! Rather, only Kidush is considered obligatory, for it was not enacted for the sake of pleasure. It is a Mitzvah like other Mitzvos, therefore one can Motzi others. The Gemara says that one can Motzi others in ha'Motzi only regarding Matzah.

Shulchan Aruch (ibid.): This is only if they do not know how to make Kidush.

Mishnah Berurah (16): This is even if they are minors.

Mishnah Berurah (20): Some argue and say that this is only l'Chatchilah; letter of the law, one can be Motzi others in any case.