Ideally, one should concentrate completely to every word that is being said
in the repetitive Amidah with no other distraction or preoccupations. I can
remember in all my years that I davened in the Minyan of the Lubavitcher
Rebbe, he always followed in the Siddur the words being said by the Chazzan
and never did anything else during that time.
However as far as your question is concerned, the B'er Haitev in Siman 124:4
quotes an opinion that states that it is not necessary to protest against
anyone that is learning or saying Tachanun [penitential prayers] during the
repeetition of the Amidah provided that they concentrate by the end of each
of the blessings to answer Amen, however there are those that disagree and
disallow it totally.
There can be a second problem which is based on what the Shulchan Oruch
writes in Siman 90:18 (O.C.) that even when something is permitted for
scholars it should still be avoided lest the regular and ignorant people
will follow suit. In the same way in our case, even though there are those
few who are able to tear themselves away from their learning every few
moments to concentrate on the ending of each of the 19 Brochos, most people
will not realize that that is what they are doing and will just learn or say
Tehillim or Tachanun without stopping to answer Amen 9and then there might not
even be enough people answering Amen).
Eliezer Wenger