Is one permitted to talk during the communal Torah reading?
Halachically Speaking Volume 3, Issue 3 discusses this. It brings many sources for further study.
See there for for more details and sources.
Consult your local Orthodox Rabbi.
The other answer already gave reasons to forbid. I'll give a broader picture as to why one might permit.
The basic gemara in question regarding speaking during leining or between one aliyah and the next is in Sotah 39a:
The bolded phrase is כיון שנפתח ספר תורה, once the sefer Torah is 'opened'. This might refer to unrolling, or to actual reading. If actual reading, then there is no prohibition even of devarim beteilim, non-Torah speech, between aliyot.
The Aruch HaShulchan (who some people follow rather than the Mishnah Berurah) writes in Orach Chaim 146 -- the same siman cited in the other answer:
In terms of studying other material during leining, there is the gemara in Berachos 8a:
The peshat in the gemara, IMHO, is just what Soncino writes:
That is, the common folk need to hear the public reading of the Torah. But we can learn at a more advanced level, and should learn gemara. This rather radical approach was at odds with expectations, as well as the statement in Sotah, that even in matters of Torah it is forbidden to speak, and so many Rishonim limited this statement in various ways. See a listing in Aruch Hashulchan. For example, that this is only once he had already fulfilled Shnayim Mikra; or only one on the caliber of Rav Sheshet; and to limit this radical approach, people tend to adopt all of the restrictions simultaneously.
Returning to the topic of #1, about speaking bein gavra legavra, consider the following:
That does not mean that one should talk. But on the other hand, it is more complicated than simply saying (as in the other answer) that:
There is a basis, and there is reason to judge others favorably.
While it proceeds to list leniencies/exceptions, the Shulchan Aruch Orach Chaim 146:2 states initially that it is prohibited even to speak Divrei Torah even in between aliya's (i.e. while the Torah reading has paused).
One leniency is for one for whom "Torah is his occupation" but the M.B. 9 cites the Elya Rabba (who cites many Rishonim) that no one today is technically included in this category.
Some permit one to study Torah quietly. Some permit one to study if there are ten people (a tzibur/congregation/minyan/etc) listening to the reading, but even then it must be done quietly (M.B. 8)
According to all opinions (M.B. 11) one may read Shnayim Mikra v'Echad b'Targum (2x in the Torah and 1x in the Aramaic translation) during the Torah reading since one is covering the same material.
There are some other issues but I see no basis to discuss other matters which are not related to Divrei Torah, even between aliyos.
Likewise the above leniencies are inapplicable to certain readings such as parshas "Zachor" or parshas "Parah" (see M.B.12, 13).
Needless to say in an area such as this where there are so many different opinions and variables one should discuss the issue with a competent Rav.