In the Gemara in tractate Berachot, on Daf 45a, a discussion arises regarding two people who eat a meal together. The central question is whether they may perform a formal invitation to bless. This issue is the subject of a dispute between Rav and Rabbi Yochanan; one sage maintains that if they wish to perform the invitation, they may do so, while the other argues they may not.
The commentary of Tosafot explores the scope of this ruling, beginning with the interpretation offered by Rashi. According to Rashi, this limitation is not restricted to the formal invitation to bless. He suggests that the same rule applies to the initial blessing over bread and, for that matter, all other initial blessings. In Rashi's view, when two people eat together, one cannot recite the blessing to fulfill the obligation for his fellow. He bases this on the statement of Abaye, who taught that when two people eat together, it is a commandment for them to separate. Rashi understands this to mean they must remain separate for all blessings, whether it is the initial blessing over bread or the grace after meals.
However, Tosafot finds this interpretation difficult to accept. He points to a discussion found earlier in the Gemara, in the chapter titled Perek Keitzad Mevarchin on Daf 39. There, the text describes an incident where Bar Kappara gave permission to one of his students to recite a blessing on behalf of another. In that case, there were only two students present, yet one recited the initial blessing for his companion. Tosafot further notes that even if one were to argue there were actually three people present, the group was eating fruit. Since the formal invitation to bless is not applicable to fruit, the case proves that an initial blessing functions differently than the formal invitation.
To support this distinction, Tosafot cites the law regarding the blessing over fire during the conclusion of the Sabbath, where it is established that one person may recite the blessing for everyone else. Therefore, Tosafot concludes that regarding initial blessings, one person can indeed fulfill the obligation for another, provided they sat down with the intention of eating together. The reason the final blessing is treated differently is practical: at the end of a meal or after eating fruit, people often finish and depart at different times. Because they naturally separate from one another as they finish, each person must recite the final blessing for themselves.
Finally, Tosafot offers an alternative resolution for why one may recite the initial blessing for another but requires a group of three for the grace after meals. This distinction may be based on the source of the obligation. Initial blessings are Rabbinic in origin, and therefore the law allows one person to fulfill the obligation for his fellow. The grace after meals, however, is a Biblical obligation. Due to the weight of this Biblical requirement, the law is more stringent, requiring a minimum of three people to be present before one individual can fulfill the obligation for the group through a formal invitation.