Answer

It is permitted to study (although there are halakhic decisors who prohibited it), since you have no role at all during the repetition of the Amidah. This is a practice whose rationale has fallen away (today we have prayer books and everyone knows how to read. True, some claimed that this enactment was not only for those who do not know how to read, but that the repetition turns the prayer into a communal prayer. But that is a post facto explanation). One should answer "Blessed is He and blessed is His name" and Amen. And of course one should stop for Kedushah and the Rabbinic Modim. The permission to read applies during the moments when you have no role to play (the dead moments). When you do have a role, you must fulfill it.
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Questioner:
By the same reasoning, would it also be permitted to study Torah during the Torah reading (on Monday, Thursday, and Sabbath)?
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Rabbi:
With regard to the Torah reading, the situation is a bit different, because opinions are divided as to whether it is an obligation of the congregation or an obligation of the individual worshippers. If it is an obligation of the congregation, then there is room to say (though even that is not necessary) that it is enough that the congregation is reading, and I can study something else. If it is an individual obligation, then it is apparently forbidden. Still, there may perhaps be room to permit listening to the reading with half an ear and the like.
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Jonathan:
You mentioned the view of those who hold that the repetition of the Amidah is the prayer of the congregation (as distinct from the silent Amidah, which is the prayer of the individual). Why do you claim that this is only a post facto explanation? Does not the fact that all Jewish communities in all places and periods practiced it strengthen that argument?
By the same token, one could even say the opposite. From the outset, the repetition of the Amidah was instituted as communal prayer, and only afterward was it explained as serving to discharge others' obligation.
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Rabbi:
Simply because there is no hint of this anywhere. There is an explicit written explanation, and I see no logic in ignoring it in favor of one speculation or another. At most, one could say that we ourselves are now re-instituting the repetition of the Amidah in order to give prayer a communal character. But then the question of authority to enact ordinances already comes into play.