The procedure of reciting Barechu was instituted by the Hachamim (sages) as a call-and-response ceremony between the Chazzan (cantor) and the rest of the Kahal (congregation). This means that the Chazzan prompts the Kahal by saying, “Barechu et Hashem ha-mevorach, let us come and bless God.” The people respond, “Baruch Hashem ha-mevorach le'olam va'ed” (Blessed is Hashem, the blessed One, forever and ever). But the Chazzan then has to go back and repeat the words, “Baruch Hashem ha-mevorach le'olam va'ed,” because at the beginning, he is merely prompting the people to bless. If the people then bless and he does not repeat it, it looks as though he is separate from them and is not participating in the blessing, which is a mistake.
Therefore, Shulchan Aruch writes emphatically in multiple places—in Siman Nun Zayin (section designation), where he talks about Barechu, as well as by Kriyat HaTorah (Torah reading), when a person gets an aliyah (Torah-reading honor)—that the one who is calling to the Kahal, “Barechu,” has to make sure to respond again, “Baruch Hashem ha-mevorach le'olam va'ed.” He must do so out loud to alert everybody that he is participating in the Beracha (blessing). He must also make sure to say Barechu out loud so that he can prompt everybody else to go ahead and to be able to say it. So a person also has to make sure that he is not saying it by muttering it to himself, but is participating together with everybody else.
There are different customs regarding whether a person needs to stand when reciting Barechu. Among the Ashkenazim, they require standing, and they all end up standing. Most Sefaradi Kehillot (communities) do not stand for Barechu. Some people you see will do a little Hiddur (enhancement) or a small Keri'ah (bowing), but it does not seem to be mandatory. A minority of Sefaradim do stand, but it is not a requirement, and they can remain seated and respond to Barechu, especially at Kriyat HaTorah.
There is one time when it seems to be more prevalent for even Sefaradim to stand, and that is on Friday night when they say Barechu, because, as we mentioned, Friday night Barechu is the acceptance of Shabbat. So, given the special Kabbalat Shabbat (acceptance of Shabbat) at Barechu, people have the custom to stand even though they do not do so the rest of the week. Have a great day.