Gemara

(Rabah bar Rav Huna): If orphans bought an Esrog with money of the estate and one of them took it for the Mitzvah, he was Yotzei only if the others would allow him to eat it.

This is only if there is an Esrog for each brother. If there are (only) other fruits for the others, he was not Yotzei.

Sukah 41b - Beraisa: "Lachem" teaches that on the first day one is not Yotzei with another's Lulav, unless he gave it for a gift.

Rishonim

The Rif (Sukah 20a) and Rosh (8:47 and Sukah 3:31) bring our Gemara.

Question: Why are the orphans Yotzei if there are other Esrogim? They did not split the inheritance!

Answer (Ran, DH v'Sham'inan): The case is, they do not mind if he takes one, therefore he is Yotzei. This is even if he took the nicest one.

Note: The Shulchan Aruch (649:5) allows to take another's Lulav without his knowledge only after the first day. Even though the owner does not mind, it is no better than a loan. We must say that partners are different, for the partners who are not using it need not be Makneh (transfer ownership of it).

Rif: We learn from Bava Basra that if two people buy a Lulav together, neither is Yotzei on the first day unless the other gives it to him for a gift.

Rambam (Hilchos Lulav 8:11): If partners bought a Lulav or Esrog, none of them are Yotzei on the first day unless the other gives to him his share for a gift. If brothers bought an Esrog with money of the estate and one of them took it for the Mitzvah, he was Yotzei only if he could eat it and the others would not mind. If they would mind, he is Yotzei only if they give him their shares for a gift.

Magid Mishnah: According to the primary text, when there is an Esrog for each brother, even if it is inferior, they are not particular, and their pardon helps. If there is not an Esrog for each brother, even if they are not adamant, their pardon does not help. They must give to him for a gift.

Magid Mishneh (citing the Rashba, and Nimukei Yosef DH Omar ha'Mechaber citing the Ritva): This is when they bought an Esrog to eat or smell. But if they bought it for the Mitzvah, they cannot divide it (that part will be for each). Rather, they intended that whoever takes it will own it at the time! This is like partners in a Chatzer who vowed not to benefit from each other. They may enter the Chatzer because (in such a case) Yesh Breirah. They divide it so that each totally owns the place he occupies at the time. This is why in many places they buy one Lulav and the entire congregation is Yotzei with it.

Rosh (Sukah 3:31): Where Esrogim are scarce, the entire Tzibur buys one in partnership and everyone is Yotzei with it. The Rashbam says that it is as if they stipulated that everyone gives his share to whoever takes it, on condition that he will return it. Rav Shrirah Gaon said that this is good after the first day, but on the first day an individual should buy it and give it to others. The Tzibur can give money as a gift to the Chazan or someone else to buy it. In Ashkenaz and France the custom is like the Rashbam.

R. Gershom (137b DH Notlo): We say that he was Yotzei only if there is an Esrog for each brother, i.e. enough money left in the estate to buy one for each.

Poskim

Shulchan Aruch (OC 658:7): If partners bought a Lulav or Esrog, neither of them is Yotzei on the first day unless the other gives to him his share for a gift.

Rema: This is when they bought an Esrog to eat or smell. If they bought it for the Mitzvah, presumably this was their intent.

Gra (DH v'Davka): We rely on Breirah, just like partners who are Mudar Hana'ah from each other. No Kinyan is needed before or after taking it.

Magen Avraham (10): In a Teshuvah, the Rashba says that he wrote this l'Halachah, but not to rely on it in practice. There are many opinions about this in Hilchos Nedarim. Therefore, they should declare that they give their shares to whoever takes it (on condition to return it).

Shulchan Aruch (8): If brothers bought Esrogim with money of the estate and one of them took it for the Mitzvah, he was Yotzei only if he could eat it and the others would not mind. If they would mind, he is Yotzei only if they give to him their shares for a gift.

Shulchan Aruch (ibid.): If each brother bought a different species, or if together they bought different species from the estate:

If one of them took it for the Mitzvah, even if he could eat it and the others would not mind, he was Yotzei only if they give to him their shares for a gift. Because there is only one Esrog, their Stam Mechilah (lack of objection) does not help.

Mishnah Berurah (36): Even though they do not mind if he eats it, they also want to be Yotzei, therefore they object to him using it for the Mitzvah.

Shulchan Aruch (ibid.): If there are other Esrogim, their Stam Mechilah helps, even if his is nicer than others.

Shulchan Aruch (9): Where Esrogim are scarce, the custom is that the entire Tzibur buys one in partnership. The reason is that since they bought it to be Yotzei, it is as if they stipulated that everyone gives his share to whoever takes it, on condition that he will return it to them.

Mishnah Berurah (38): Our custom is that everyone gives as much as he wants. It seems that one who could have given, but did not give anything is not Yotzei.

Rema: We collect for it according to people's wealth, for Hidur Mitzvah is incumbent more on the wealthy. Everyone should be zealous to get his own four Minim to fulfill the Mitzvah properly.

Darchei Moshe (4): If one has Yir'as Shomayim and cherishes Mitzvos he will get his own Minim to shake them properly (at the proper times in Hallel). Also, most people think that one may bless on a jointly owned Esrog. The Maharil says that even when the first day is Shabbos one should strive to have his own Minim.

Mishnah Berurah (39): It is best to take one's own Minim, even if they are inferior to his friend's, for not everyone knows how to be Makneh properly. Taking a friend's is better than taking the Tzibur's, for perhaps someone in the Tzibur did not intend to Makneh his share to the others.

Kaf ha'Chayim (81): The SHLaH says that it is better to bless on the Tzibur's Minim than on those of a friend given on condition to return, for the Tzibur's is called "Lachem". This is unlike all the Poskim.

Kaf ha'Chayim (89): Some say that to fulfill "Lachem" absolutely, one should not give his Lulav to another before he himself blesses on it. Others disagree, for after the gift is returned it is fully his.

Kaf ha'Chayim (19): If a Tzibur elsewhere has no Minim, it is better to give one's Minim to them and to perform the Mitzvah with those of his Tzibur. Some say that one should not do so to enable an individual to fulfill the Mitzvah, especially if the Tzibur's Minim become Pasul through great usage after a few days.