Before us is Mishnah 7 in the first chapter of Maseches Shabbos. In this Mishnah, Beis Shammai and Beis Hillel disagree, and the dispute does not deal with the resting of vessels as in the previous Mishnayos, but with a related idea.
The words of Beis Shammai:
"Beis Shammai omrim: ein mochrin l'nochri, v'ein to'anin imo, v'ein magbihin alav, ela k'dei sheyagia l'makom karov" - three things are forbidden here on Friday afternoon:
"Ein mochrin l'nochri" - one may not sell an object to a non-Jew.
"V'ein to'anin imo" - one may not help him load a burden onto a donkey.
"V'ein magbihin alav" - one may not assist him in placing the burden onto his own back.
"Ela k'dei sheyagia l'makom karov" - all of this applies unless the non-Jew's destination is close by, such that he can reach it before Shabbos begins.
The reason for the prohibition is because of appearances: if the non-Jew sets out on a long journey, it will look to onlookers as though he is taking the object as the Jew's agent and carrying it on his behalf on Shabbos. This is the concern that troubled Beis Shammai.
The words of Beis Hillel:
"U'Beis Hillel matirin" - Beis Hillel permitted selling the object to a non-Jew on Friday afternoon, helping him load the donkey and placing the burden on his back, and even when the destination is a distant place that he will not reach until after dark.
However, even according to Beis Hillel, the Gemara notes that the non-Jew must leave the Jew's house before Shabbos begins, for otherwise it appears as though the object was sold to him on Shabbos itself.
In summary: in this Mishnah, Beis Shammai and Beis Hillel disagree about three acts done together with a non-Jew on Friday afternoon: selling to him, helping him load, and lifting a burden onto him. Beis Shammai forbid these out of concern for appearances, that it will look as though the non-Jew is carrying the burden as the Jew's agent on Shabbos, and they permit it only when the destination is close and he is able to reach it before Shabbos. Beis Hillel permit it even to a distant place, provided that the non-Jew leaves the Jew's house before Shabbos begins.