The Shulhan Aruch (Orah Haim 66) writes that if one hears Kaddish while reciting the first two verses of Shema – "Shema Yisrael" and "Baruch Shem" – he may not respond to Kaddish. If he hears Kaddish after he has recited the first two verses, as he recites any part of the rest of Shema, then he joins the first five "Amen" responses, even though he is in the middle of reciting Shema. These include the response of "Amen Yeheh Shemeh Rabba." Hacham Ovadia Yosef rules that in such a case, one responds only until the words "U’le’olmeh Almaya Yitbarach," without continuing until "Da’amiran Be’alma." Our custom, however, follows the opinion of the Kaf Ha’haim (Rav Yaakov Haim Sofer, Baghdad-Jerusalem, 1870-1939) and Hacham Bension Abba Shaul (Jerusalem, 1924-1998), that even while reciting Shema, one responds "Amen Yeheh Shemeh Rabba" all the way through "Da’amiran Be’alma."
Likewise, during the Shema recitation, after the first two verses, one who hears "Barechu" responds "Baruch Hashem Ha’meborach Le’olam Va’ed" together with the congregation. One who hears "Modim" while reciting Shema (after the first two verses) responds with only the first three words – "Modim Anahnu Lach." One does not answer "Amen" to a Beracha that he hears while reciting Shema, except to a Beracha which is recited only in the presence of a Minyan. Namely, he answers "Amen" to the Beracha recited by the Kohanim before Birkat Kohanim ("Asher Kideshanu Bi’kdushato Shel Aharon…"), to all three verses of Birkat Kohanim itself, and to the Berachot recited over the Torah reading, both the Beracha recited before the reading ("Asher Bahar Banu…") and the Beracha recited after ("Asher Natan Lanu Torato…"). All these Berachot are recited only when a Minyan is present, and so one answers "Amen" to these Berachot even if he is reciting Shema (except during the first two verses).
Hacham David Yosef (contemporary), in his Halacha Berura, writes that these guidelines apply only when one recites Shema for the Misva, to fulfill the obligation to recite Shema each morning and evening. If, however, one is not fulfilling the Misva through his recitation, then he answers "Amen" to every Beracha, and joins in all "Amen" responses to Kaddish, without any restrictions. Hacham David gives numerous examples of situations where one recites Shema but is not fulfilling the Misva. One example is one who puts on Tefillin Rabbenu Tam each morning after the prayer service, and recites Shema while wearing the Tefillin. Other examples include one who prays Shaharit or Arbit but had already recited Shema earlier; one who recites Arbit early, before dark (as many communities do on Friday night during the summer months), and must repeat Shema after dark to fulfill the Misva; and one who recites the bedtime Shema. In all these situations, the individual recites Shema without fulfilling the Misva. As such, Hacham David writes, he interrupts for any response, and to answer "Amen" to any Beracha, even while reciting the verses of "Shema Yisrael" and "Baruch Shem."
Although Hacham Bension Abba Shaul held a different view, Hacham David maintains – and this appears to have been the view of his father, Hacham Ovadia Yosef – that one may interrupt Shema for any response of "Amen" and the like, at any point, in a case where he is reading Shema without fulfilling the Misva.
Summary: While reciting the first two verses of Shema – "Shema Yisrael" and "Baruch Shem" – one may not interrupt, even to respond to Kaddish and the like. After the first two verses, however, one who hears Kaddish joins in the first five "Amen" responses to Kaddish, including "Amen Yeheh Shemeh Rabba," and he recites through "Da’amiran Be’alma" as usual. He also responds to "Barechu," and if he hears "Modim," he responds with the words "Modim Anahnu Lach." He does not answer "Amen" to Berachot, except to the Berachot before Birkat Kohanim, the blessings of Birkat Kohanim itself, and the Berachot before and after the Torah reading. These guidelines apply only when one recites Shema for the Misva. If, however, one is not fulfilling the Misva through his recitation – such as if he had already recited Shema, or if he recites Arbit early, before dark, and will repeat it later – then he responds "Amen" to all Berachot, and joins all responses to Kaddish, without any restrictions, even if he is reciting the first two verses of Shema.