Is one permitted to talk during the communal Torah reading?
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.
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:
Raba son of R. Huna said: When the Torah-scroll is unrolled it is forbidden to converse even on matters concerning the law; as it is said: 'And when he opened it all the people stood up', and standing up signifies nothing else than silence, as it is said: 'And I wait because they speak not, because they stand still and answer no more.' R. Zera said in the name of R. Hisda: [It may be derived] from this passage, 'And the ears of all the people were attentive unto the book of the law.'
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:
ודע שזה שאמרה הגמרא "כיון שנפתח ספר תורה אסור לדבר" – הכוונה משעה שמתחילין לקרוא. וכן כתב הרמב"ם בפרק שנים עשר. ויש שרוצה לומר דמשעה שנפתחה, גם קודם קריאתה – אסור (עיין מגן אברהם סעיף קטן ג'). ואינו כן, דלהדיא פירש רש"י שם: כיון שהתחיל לקרות, עיין שם. וכן פסקו הגדולים (מעיי"ט ואליה רבה סעיף קטן ג').ש
ודע דרבינו הבית יוסף כתב דאסור לספר אפילו בין גברא לגברא. ויש מתירין בין גברא לגברא, ובפרט האידנא שמאריכין ב"מי שבירך" (ב"ח ומגן אברהם). ובאמת גם כוונת רבינו הבית יוסף הוא מטעם דאתי לאמשוכי (אליה רבה), ולכן עניין קצר – מותר. ובכל זה אין חילוק בין שכבר קרא "שנים מקרא ואחד תרגום" או לא (בית יוסף).ש
And know that this that the gemara stated 'once the sefer Torah is opened it is forbidden to speak' -- the intent is from the time they begin to read. And so writes the Rambam in perek Shnaim Asae. And there is one who wants to say that it is from the time it is unrolled, even before reading, it is forbidden (see the Magen Avraham seif karan 3). And it is not so, for explicitly does Rashi explicitly explains there -- once he begins to read, see there. And so pasken the Gedolim.
And know that Rabeinu the Beis Tosef wrote that it is forbidden to speak even between one aliyah and the next. And there are those who permit between one aliyah and the next, and specifically nowadays that they extend in Mi Shebeirachs (see Bach and Magen Avraham). And in truth even the reason of the Beit Yosef is because he will come to extend (Eliah Rabba), and therefore, a short matter is permitted. And in all this there is no difference between whether he read Shnayim Mikra or not.
In terms of studying other material during leining, there is the gemara in Berachos 8a:
R. Shesheth used to turn his face to another side and study. He said: We [are busy] with ours, and they [are busy] with theirs.
The peshat in the gemara, IMHO, is just what Soncino writes:
They are engaged in listening to the public reading and we, more profitably, with more advanced study.
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:
28 Machatzis ha-Shekel, Aruch ha-Shulchan, and Shulchan ha-Tahor maintain that the Bach permits even idle talk between aliyos. See also Pri Chadash who permits conversing bein gavra l'gavra. Obviously, they refer to the type of talk which is permitted in shul and on Shabbos.
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:
I see no basis to discuss other matters which are not related to Divrei Torah, even between aliyos.
There is a basis, and there is reason to judge others favorably.
Halachically Speaking Volume 3, Issue 3 discusses this. It brings many sources for further study.
Once the sefer Torah is opened it is forbidden to talk - even words of Torah.
One is not allowed to speak between Aliyot, since he might end up missing part of the reading.
Shnayim Mikra during the Torah reading is permitted, but not recommended.
The poskim say regarding one who talks during the Torah reading: “what will he answer on the Day of Judgment?”
See there for for more details and sources.