The custom is when someone is being Yotzei in a Mitzva from someone else (such as hearing the Shofar or reading of the Megilla), the person listening does not answer 'Boruch Hu Uvoruch Shemo' to the one making the Brocha. I assume the reason is that this would constitute a Hefsek - interruption between the Brocha & the Mitzva the Brocha is going on. However, I have noticed people who refrain from saying 'Boruch Hu Uvoruch Shemo' during the Chazzan's repetition of Shemona Esrei. What would be the reason for this? The Congregation already Davened their own Shemona Esrei so they aren't trying to be Yotzei with the Shliach Tzibur?
Discussing the customs of Rabbi J B Soloveitchik it says:
During Chazaras HaShatz (…the cantor’s repetition of the Amidah) he would never answer ‘Baruch Hu Avaruch Shemo’ (…blessed is He and blessed is His name) as is the common practice and which is mentioned in the Shulchan Aruch.. He held that the cantor’s Amidah was a congregational (Tzibur) requirement and as such no interruptions may be made if we are to fulfill our obligation. This is the Halacha by Brachos. If one is to fulfill the requirement of the Bracha, one may not interrupt the Bracha by saying ‘Baruch Hu Avaruch Shemo’. One must only answer Amen at the end of each Bracha.
So the followers of "The Rav" presumably follow his custom in this.
The Mishna Brura on 124:5 says that the chazan needs to wait for the people to finish saying baruch hu uvarch shemo before finishing the bracha so that the people can say amen to the bracha. One concern is the chazan will be unlearned in hilchot shaliach tzibbur and will finish the bracha while everyone is saying Baruch hu Uvaruch Shemo and therefore there are people who forgo saying Baruch Hu Uvaruch Shemo so that they can make sure to be able to say amen.
The Gra did not recite ברוך הוא וברוך שמו because it may prevent answering amen (Maaseh Rav, 43) or because he considered it to be a hefsek (Tosefes Maaseh Rav). See also Earliest source for "Baruch Hu U'Baruch Sh'mo".