Eruvin, chapter 6, mishnah 3. The mishnah deals with "anshei chatzer" - the residents of a courtyard, each of whom has a house opening into it - where one of them forgot to join in the eruv. The mishnah itself does not say so explicitly, but the Gemara explains that the case is one in which he then renounced his rights in the courtyard.
The basic principle: a person who did not join in the eruv may, on Shabbos, give over his rights in the courtyard to the others, so that they will be able to carry in it. He himself, however, may not carry from his house into the courtyard.
"Sheshachach echad mehen v'lo eirev" - his house is forbidden to him and to them:
Even though he handed over his rights in the courtyard, carrying from his house is forbidden - to himself certainly, since by doing so he would be retracting the renunciation he made, but also to them: they too may not carry from his house into the courtyard. The reason is that the tanna of our mishnah holds that when a person renounces his rights without specifying, we assume only that he renounced the rights he has in the courtyard, and not the control he has over his house, unless he said so explicitly. Therefore his house remains forbidden for carrying, both to him and to them.
"V'shelahem mutarin lo v'lahem":
In that very same case, where he renounced his rights in the courtyard, carrying from their houses into the courtyard is permitted to both parties:
To them - since they received his renunciation of rights, and are therefore permitted to use the courtyard and to carry from their houses into it.
To him - he too may carry from their houses into the courtyard, since he is considered a guest in their homes.
"Nasnu lo reshusan - hu mutar v'hen asurin":
Now the mishnah speaks of the reverse case: the one who forgot to join in the eruv did not renounce his rights; rather, the other residents of the courtyard renounced their rights to him. In this case he is permitted to carry from his house into the courtyard, since they renounced their rights to him, and they are forbidden to carry from their houses into the courtyard, since they renounced their rights in it. The halachah is that they may not bring things out into the courtyard even from his house.
One might ask: was it not stated above that when he renounced his rights, he was able to bring things out from their houses into the courtyard by virtue of the law of a guest? The answer is that one person can be considered a guest of a group of others, but we do not say that many people are considered guests in the house of a single person. The reverse direction does not work.
"Hayu shnayim - osrin zeh al zeh":
What if two people forgot to join in the eruv, and the rest of the residents of the courtyard, who did make an eruv, said to them: we will renounce our rights to you, so that you will be able to carry in the courtyard? The mishnah rules that they render it forbidden for one another, meaning that this renunciation is of no help. The reason: since each of the two retains his own rights, and they themselves neither made an eruv nor renounced their rights, their rights render the courtyard forbidden to each other - and they are like two people in a courtyard who neither made an eruv nor renounced their rights. Therefore, even though the others who made the eruv renounced their rights to them, their renunciation is of no help.
The Gemara adds that even if afterward one of these two renounces his rights to his fellow, it does not help: at the time when those who made the eruv renounced their rights to them, the renunciation had no validity, and therefore it remains without validity.
In summary: the mishnah concludes with the general rule:
An individual - gives over his rights to the rest of the residents of the courtyard with whom he did not make an eruv, and he may also receive rights from them if they renounced their rights to him.
Two - they give over rights, and they can renounce their rights to the rest of the residents of the courtyard who made an eruv, and such a renunciation is effective; but they do not receive rights, and they cannot accept rights from those who made the eruv and wish to renounce their rights to them.
And the reason, as explained: since the recipients neither made an eruv nor renounced their rights, they render it forbidden for one another. Their respective rights, their ability to carry in the courtyard, and the control they have over it, cause each of them to render it forbidden for his fellow, and they remain like two people who did not make an eruv. Therefore, receiving rights from the others does not help them at all.