During Kaddish, the Hazzan must recite "Yeheh Shemeh Rabba" along with the congregation. The Mishna Berura writes that the Hazzan recites "Yeheh Shemeh Rabba" silently, and then raises his voice again when he reaches the word "Yitbarach." The work Az Nidberu (Rav Binyamin Zilber, 1916-2008) explains the Mishna Berura to mean that the Hazzan may recite "Yeheh Shemeh Rabba" silently if he so wishes, but he must then raise his voice beginning with the word "Yitbarach." This ruling, the Az Nidberu explains, is based on the Lebush (Rav Mordechai Yoffe, 1530-1612), who writes that when the congregation responds to Barechu by declaring, "Baruch Hashem Ha'meborach Le'olam Va'ed," the Hazzan also makes this declaration, and may do so silently if he so wishes. By the same token, the Hazzan may recite "Yeheh Shemeh Rabba" silently.
It is unclear, however, why, according to this view, the Gemara's promise of reward for one who responds to "Yeheh Shemeh" with "all his strength" ("Be'chol Koho") does not apply to the Hazzan. We would assume that just as the members of the congregation are encouraged to respond "with all their strength," this should be true of the Hazzan, as well. The answer, as some have suggested, might be that the "Yeheh Shemeh Rabba" response is primarily for the congregation, and not for the Hazzan. The Hazzan joins only so that he does not exclude himself from the congregation. Therefore, it is not necessary for him to recite it loudly.
However, there seems to be a different reason to require the Hazzan to recite "Yeheh Shemeh Rabba" aloud. The Shulhan Aruch (104:7) writes that if somebody hears Nakdishach or Kaddish while he recites the Amida, he should pause and listen attentively to the Hazzan, whereby he can be considered to have responded to Nakdishach and Kaddish. This ruling clearly assumes that the Hazzan recites the entire Kaddish – including "Yeheh Shemeh Rabba" – loudly, such that the congregants who are still reciting the Amida can listen and thereby fulfill the Misva of responding. It is possible that the Mishna Berura referred to a case where nobody in the congregation was still reciting the Amida, and for this reason, he wrote that the Hazzan does not need to recite "Yeheh Shemeh Rabba" out loud.
Regardless, the accepted practice is that the Hazzan recites "Yeheh Shemeh Rabba" aloud, just like the rest of the Kaddish.
In light of what we have seen, the Hazzan must have in mind when reciting Kaddish (and Nakdishach, for that matter) that his recitation should fulfill the obligation for those who are reciting the Amida and thus cannot respond. Those who wish to fulfill the Misva by listening to the Hazzan can do so only if both they and the Hazzan have this in mind, and it is therefore imperative that the Hazzan has in mind to fulfill the obligation for those who cannot respond. In fact, the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that if a person reciting the Amida knows that the Hazzan is unaware of this Halacha, and does not have in mind for his Kaddish recitation to fulfill the obligation for those reciting the Amida, then there is no reason for this person to interrupt his Amida prayer to hear the Kaddish. Since the Hazzan does not have in mind for his recitation to fulfill the listeners' obligation, the listeners have no possibility of fulfilling their obligation, and they might as well just continue their Amida without pausing to hear the Hazzan's recitation of Kaddish. Hacham Ovadia Yosef concurred with this ruling.
Summary: During Kaddish, the Hazzan recites "Yeheh Shemeh Rabba" together with the congregation. He should do so in a loud voice – like the rest of Kaddish – and should have in mind that those who are reciting the Amida, and thus cannot respond, will fulfill their obligation by listening to his recitation.