The source for saying “Baruch hu u’varuch sh’mo” is the Tur, where the author (Rav Yaakov Baal HaTurim) quotes his father, the Rosh:

I heard my father and teacher say “Baruch hu u’varuch sh’mo” each and every time that he heard a blessing. This fulfills what Moses said, “When I say the name of Hashem, give glory to our God” (Devarim 32:3). Furthermore, even when one mentions a righteous human being [who has passed away], one should bless him by saying, “May his memory be a blessing.” (Tur, Orach Chaim 124).

There are those who refrain from saying “baruch hu u’varuch sh’mo” (Vilna Gaon). This is because they are concerned, either that they may miss hearing the conclusion of the blessing, or that they may be breaking up the blessing.

A person who is listening to someone else’s blessing in order to fulfill (“be yotze”) his own obligation should not say “baruch hu u’varuch sh’mo.” Doing so in this context would be a hefsek (disruption) of the blessing.

The source for “amen” is in the Torah portions of Naso and Ki Tavo. In a few places, the Mishnah discusses saying “amen” after blessings. There are two primary meanings of “amen,” and when responding “amen” one should have in mind the meaning appropriate to the blessing:

1) “I wish.” The respondent hopes for the realization of the blessing. One should intend this when responding to blessings which are requests, such as the “amen” said after Kaddish.

2) “I believe in what was said in this blessing.” One should intend this when responding to blessings of praise, thanks, and enjoyment. For example: I believe that God brings forth bread from the ground (motzi lechem min ha’aretz). In the Gemara (Shabbat 119b), Rabbi Chanina explains that “amen” (aleph-mem-nun) is an acronym for “E-l melekh ne’eman (God, faithful King).”

There are some blessings where both meanings are apt, such as “Sim Shalom” (May You Grant Peace). We both believe and hope that God will bless His nation with peace.

One should respond to all blessings with “amen.” This includes blessings to which he is listening in order to fulfill his own obligation. The reason is that unlike “baruch hu u’varuch sh’mo,” with “amen” there is no problem of hefsek (since the response is at the end of the blessing).

Our Sages praise those who respond “amen.” A number of Tannaim are of the opinion that the one who responds “amen” to a blessing is greater than the one who actually makes the blessing: “Know that while the common soldiers advance and open the battle, it is the seasoned warriors who go down to win the victory” (Berakhot 53b). The one who finishes the job gets the credit.

Responding “amen” with intent gives voice to the belief of the one praying. The Arukh HaShulchan (Orach Chaim 124:11) writes that the foundation of faith hangs on “amen,”and therefore one should train his children to always respond “amen.”