If a group of people are reciting a Zimun, and there is somebody sitting with them who did not eat, he does not answer with the standard response of "Baruch She’achalnu Mi’shelo…" Since he did not eat, he cannot say "She’achalnu Mi’shelo" ("from whom we ate"). Instead, the Shulhan Aruch (Orah Haim 198) writes, he responds, "Baruch U’meborach Shemo Tamid Le’olam Va’ed." (An mnemonic to help remember this response is "Boshet Lo.") If ten people are present, such that those who ate answer with Hashem’s Name ("Baruch Elokenu She’achalnu Mi’shelo…"), a person who did not eat responds, "Baruch Elokenu U’meborach Shemo…" And if this takes place at a Sheba Berachot, then the person who did not eat responds, "Baruch Elokenu U’meborach Shemo Tamid Le’olam Va’ed She’ha’simha Bi’m’ono."
However, if a person ate or drank anything – then even in circumstances where he would not be counted for a Zimun, he nevertheless recites the standard text of "Baruch She’achalnu Mi’shelo…" For example, if three people are making a Zimun, and there is a fourth person present who did not eat anything but drank water, he answers "Baruch She’achalnu Mi’shelo…" along with the others. Even though he would not be counted for the Zimun if there were only two other people besides him – since he only drank water, and did not eat anything or drink wine – nevertheless, his having drunk water suffices for him to be able to say, "Baruch She’achalnu Mi’shelo…" Similarly, if a fourth person joined the group and ate after the others had washed Mayim Aharonim in preparation for Birkat Ha’mazon, he answers, "Baruch She’achalnu Mi’shelo…" Although he would not be counted for a Zimun if he were the third person – since he arrived after the others washed Mayim Aharonim – he nevertheless responds, "Baruch She’achalnu Mi’shelo…" since he ate. In these cases, the person can say "She’achalnu" like the others even though he would not be counted as one of the three to facilitate a Zimun.
If somebody arrives just as the people in the group are reciting, "Baruch She’achalnu Mi’shelo," and he did not hear the Mezamen (the one leading the Zimun) first announce, "Nebarech She’achalnu Mi’shelo," then he simply recites "Amen" to their response. Their response of "Baruch She’achalnu" is no different than a regular Beracha, and so this person who heard their response answers "Amen." He does not answer "Amen" again after the Mezamen repeats "Baruch She’achalnu Mi’shelo…" If he arrives just as the Mezamen repeats "Baruch She’achalnu Mi’shelo…" then he answers "Amen" to the Mezamen’s response.
Summary: If somebody is present when a Zimun is being recited, but he had not eaten anything, then he responds, "Baruch U’meborach Shemo Tamid Le’olam Va’ed," instead of the standard, "Baruch She’achalnu Mi’shelo..." If a Minyan is reciting the Zimun, then he responds, "Baruch Elokenu U’meborach Shemo Tamid Le’olam Va’ed." However, if he ate or drank anything, even water, and even if he arrived and ate after the others had washed Mayim Aharonim, then he answers with the standard response of "Baruch She’achalnu." One who did not hear the leader of the Zimun announce, "Nebarech She’achalnu Mi’shelo," and heard only the group’s response of "Baruch She’achalnu…" answers "Amen" to their response, and then does not answer "Amen" to the leader’s repetition of "Baruch She’achalnu Mi’shelo…"
