The Gemara in Masechet Shabbat (119) speaks of the great merit earned by responding "Yeheh Shemeh Rabba" to Kaddish, stating that responding "with all one's strength" has the ability to annul harsh decrees issued against a person.
The conventional understanding of this teaching is that it refers to responding with Kavana (concentration), with full attention and emotion. However, Tosafot cite an explanation that the Gemara speaks here of responding in a loud voice. Accordingly, the Shulhan Aruch writes that one should respond to Kaddish both with Kavana, and in a loud voice.
Although it is proper to respond to Kaddish out loud, one must ensure not to turn his response into a spectacle. The purpose of Kaddish is to bring glory to Hashem – not to bring glory to oneself. Therefore, if one responds to Kaddish in a manner that brings attention to himself, this undermines the entire purpose of this prayer.
Generally speaking, it is improper when responding to the Hazzan's prayer to do so more loudly than the Hazzan. The verse in Tehillim (34:4) states, "Gadelu L'Hashem Iti" – "Give praise to G-d with me," implying that the leader and the congregation should pray together, at the same volume, without trying to drown out each other. Accordingly, when responding to Kaddish, one should ensure not to respond in a louder voice than the Hazzan (or of whoever it is who recites Kaddish).
Interestingly, Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995) maintained that this Halacha does not apply to the response of "Yeheh Shemeh Rabba." He contended that "Yeheh Shemeh Rabba" is not a response to the Hazzan, but rather an independent declaration. Therefore, one may recite "Yeheh Shemeh Rabba" even more loudly than the Hazzan.
The accepted Halacha, however, does not follow this view, and instead treats "Yeheh Shemeh Rabba" as a response, which must therefore not be declared in a louder voice than the Hazzan's. Accordingly, the Kaf Ha'haim (Rav Yaakob Haim Sofer, Baghdad-Jerusalem, 1870-1939) writes that the Hazzan or person reciting Kaddish should ensure to say Kaddish loudly, so that the congregation can answer out loud without answering more loudly than him.
Summary: One should respond to Kaddish with concentration and in a loud voice, though one should not respond more loudly than the Hazzan (or other person reciting Kaddish). Therefore, one who recites Kaddish should do so loudly, so that the congregation can respond loudly but not more loudly than his recitation.