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Answer: The first halachic code that mentions the phrase Baruch hu u’varuch shemo (meaning that we "bless" Hashem and His Name) is the Tur in two places. It is not found in the gemara or the halachic works based on it (Rambam, Rif, Rosh). In Orach Chayim 124, the Tur cites an oral statement of his father (the Rosh) to recite the phrase upon hearing all berachot, in line with the statement that Moshe taught Bnei Yisrael to praise Hashem whenever he mentioned His Name (Yoma 37a based on Devarim 32:3). The Shulchan Aruch (OC 124:5) brings this as the halacha. (It is not a full obligation, and therefore it should not be said when it would harm a beracha - Mishna Berura 124:22.)
The second place the Tur mentions Baruch hu u’varuch shemo is regarding zimun (OC 192), as part of his text at the end of the mezamen’s final recitation. The Maharshal (see Taz 192:1) considers it a misprint, and Rav Yosef Karo ignores it in both the Beit Yosef and the Shulchan Aruch. However, other of the Tur’s commentaries (Bach and Perisha) find earlier sources (Rokeach and Avudrohom (with a different text)).
We found three explanations for the rationale to recite Baruch hu u’varuch shemo in zimun. The Bach views it as an extension of the Rosh/Tur’s idea of blessing Hashem upon hearing His Name in a beracha. Therefore, he reasons, it applies only in a zimun of ten, when His Name (i.e., Elokeinu) is used. The Bach adds that this formulation is particularly appropriate here because the same pasuk (Devarim 32:3) is a source for saying Baruch hu u’varuch shemo and for the requirement of zimun in general (Berachot 45a).
The Darchei Moshe (OC 192:2) posits that the Tur intended that it create a desirable break between the zimun and Birkat Hamazon (the basic idea and different opinions about a short recitation between beracha groups is found in Shulchan Aruch and Rama, OC 215:1). The Eliya Rabba (192:2, see also Pri Megadim 192, MZ 1) connects this with a minhag which very few people practice today – that the zimun responders answer Amen to the mezamen’s "Baruch she’achalnu …" (see opinions in Magen Avraham, introduction to siman 192). Amen is their break; the mezamen’s break is Baruch hu u’varuch shemo. According to this, since we do not answer Amen, the responders might want to say Baruch hu u’varuch shemo as well.
The Perisha (OC 192:2) and the Maharal (Netivot Olam, Netiv Ha’avoda 18) connect Baruch hu u’varuch shemo to the idea of adding on to one’s counterpart’s blessing (see Taz, Yoreh Deah 242:5). Here, every time the response switches sides, something should be added – the responders add "… u’v’tuvo chayinu"; the mezamen adds "Baruch hu u’varuch shemo." If so, of course it would be only the mezamen who recites it, as it sounds from the language of the Tur.
In addition to the Shulchan Aruch not bringing the minhag of saying Baruch hu u’varuch shemo, the Rama (despite his suggested explanation in Darchei Moshe) does not believe it is worthwhile, nor does the Taz (OC 192:1). The Magen Avraham (ibid.) and the Mishna Berura (192:4) cite both the practice of saying and of not saying, and view the former as more prevalent. It is hard for me to say which is more common today. For Sephardim, the Yalkut Yosef does not mention Baruch hu u’varuch shemo regarding zimun. The Kaf Hachayim (OC 192:8) discusses the counter indications and says that due to lack of clarity, it is better to refrain. The reason to not welcome additional nice words could be out of opposition to post-Talmudic additions. It is also possible that, between zimun and Birkat Hamazon, it is forbidden to break for unnecessary things (see dilemma of K’tzot Hashulchan 45:(35)).
In conclusion, if one does not have a minhag one way or another, the stronger option is to not recite Baruch hu u’varuch shemo, at least if there is there are not ten for the zimun.
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