Public policy matters, especially concerning the way tefilla operates, are the domain of the congregational rabbi. In this response, we assume that your shul either does not have a rav or that you want to know whether or how to bring the matter to his attention.
The Rosh[1] strongly opposed those who recite chazarat hashatz along with the chazan, for several reasons. Most of his concerns do not apply (or are less applicable) to the present case, but one main potential issue may remain. Let us address our case in view of the Rosh’s issues.[2]
Issue 1 – If one says chazarat hashatz along with the chazan, he is making berachot l’vatala,[3] as he has already said his private Shemoneh Esrei. In our case, though, congregants recite only sections or words that do not include the “baruch ata HaShem…” formula, which is the problematic portion of the beracha.[4] The concern that the congregation might continue on to the beracha’s conclusion[5] does not apply, assuming that the parts sung in your shul never extend that far.
Issue 2 – By singing along, one does not have the opportunity to answer “amen,” as he is not allowed to utter this response right after reciting the same beracha.[6] This, too, does not apply in our case for the above reason.
Issue 3 – It is arrogant[7] and frivolous to sing along out loud with any part of the repetition. This applies when a self-appointed assistant accompanies the chazan. However, when the congregation finds it uplifting to sing some sections together, it is not necessarily arrogant or frivolous.
Issue 4 – The Mishna Berura[8] and Igrot Moshe[9] maintain that the requirement that nine people listen to chazarat hashatz[10] applies not only to the end of each beracha, but to its entirety.[11] One might want to argue that the principle of shomei’a k’oneh allows one to combine the parts of chazarat hashatz that he hears from the chazan with those he recites himself into one complete unit. In fact, when a chazan is unable to continue, we allow someone else to continue without going back to the beginning;[12] thus, one can fulfill chazarat hashatz (b’di’eved) with multiple chazanim. However, that applies only in between berachot. If a chazan is replaced in the middle of a beracha, the new one must start at the beginning of that beracha,[13] even if he had been listening to every word until that point.[14] Thus, two cannot share one beracha. Furthermore, it is difficult to concentrate on words that a group recites in unison.[15]

This matter can be addressed as follows: Firstly, when only a few words are sung in unison, the words that are not heard properly usually do not disqualify the beracha.[16] Even in critical sections, if the congregation only provides some background voices, then there will be at least nine who hear the chazan clearly enough to fulfill the requirements of chazarat hashatz. When the congregation totally drowns out the chazan during entire sections of the tefilla, it is proper for him to wait until quiet is restored before reciting that section.
Let us put things in perspective: Although, from a purist’s approach, it is best for everyone to listen silently to the chazan with great concentration, we should be realistic. Practically, in most of our shuls, singing together significantly enhances the atmosphere and increases concentration. Therefore, trying to prevent it may not only be unfeasible, it would probably undermine efforts to achieve the atmosphere necessary to keep our shuls inviting, vibrant, and focused.