The Shulhan Aruch (Orah Haim 124:7) rules that idle chatter is forbidden during the repetition of the Amida and says that one who speaks is considered to be gravely sinning and is worthy of rebuke. The Shulhan Aruch rarely uses such harsh language and in this case is quoting Rabbenu Yonah, who in turn was quoting the Zohar’s explanation of the severity of speaking in the synagogue. The Zohar (Parashat Teruma) says that one who (flagrantly and consistently, it seems) talks during the repetition is forfeiting one’s portion among the Jewish people as well as forfeiting a resting place for one’s soul, G-d forbid.
Rabbi Haim Palagi (Tochahot Haim, Parashat Teruma) explains that in our current exile, the synagogue functions as a “Mikdash Me’at”, a miniature Bet HaMikdash and is equal to it in sanctity. Thus, just as one would not speak in the Holy Temple, one should refrain from speaking in the synagogue as well. Furthermore he explains that one who is careful not speak idly in the synagogue will merit to have children, a long life, success in one’s endeavors and other blessings.
Summary: It is forbidden to speak unnecessarily during Hazarat HaShatz. The reward is significant for one who is mindful not speak during the Hazara, or more generally, in the synagogue.