Argthtjtr

Unfortunately, there are many shuls where the majority of people talk during the service. Even more unfortunate, is that some of the people who do the most talking are primary synagogue officers with the power to hire and fire the rav. I've seen it happen a few too many times when the rav has asked these people to keep quiet, and they get offended because they don't want anyone telling them what to do. Eventually, they had the rabbi fired.

In some cases, the rav, himself, also talks during parts of the davening, about matters unrelated to the needs of the service.

Someone in my neighborhood left the shul where the rav was in both positions - i.e., he talked, and because he was afraid to lose his job, he let the people talk as well. One day, when the rav saw this person on the street, he asked him why he left the shul. The person was embarrassed to reproach the rav, because he respected him. So, he avoided the question, altogether. He asked me for suggestions, and I'm not sure about what to do.

What's the best way to reproach the rav? Tell him the blatant truth including the fact that he is part of the problem - namely that he talks? Encourage the rav to tell the main people that they cannot talk when doing so may risk his job, there? Or, continue avoiding the question and mind one's own business?

There seem to be two questions here. One is the title question: how to reproach a Rav. The implied question is how to stop talking in shul. (I have not tried to answer the implied question.)

The general answer to the title question is found as @mbloch's answer to Is there anything wrong with correcting your Rabbi in front of other people?.

Rabbi Kaganoff has an article that touches specifically on talking during the chazaras hashatz.
The implication of his article is that generally reproach should be in private.

Admonishing a talmid chacham

If someone who is not scholarly sees a talmid chacham do something
that appears to be halachically incorrect, what is the proper thing
for him to do? Does the non-scholarly person have a mitzvah to
admonish the Torah scholar for his lapse?

The halacha is that one is required to rebuke the talmid chacham, and
that even a disciple has a responsibility to be mochiach his own rebbe
(Bava Metzia 31a). There are halachic details for giving such
tochacha.

The easiest approach is for the student to ask his rebbe respectfully
what is the halacha in the situation (that was ostensibly violated).
In this way, the disciple neither acts nor speaks disrespectfully
since he did not tell his rebbe that he had committed a violation. If,
indeed, the rebbe was in violation of a halacha, it has now been
brought to his attention in an appropriate way. It also may be true
that the rebbe is aware of opinions who permit the action under the
specific circumstances involved.

The Gemara (Shabbos 55a) provides an example of this: Rav Yehudah was
listening to the Torah lecture of his rebbe, the great amora Shmuel,
when a woman entered and began screaming at Shmuel. Shmuel ignored the
woman and continued his teaching. Rav Yehudah turned to his master,
asking him: Does the master not accept the teaching of Mishlei
(21:13): “One who closes his ears from the outcry of the poor will not
be answered when he calls out (in prayer).” If Shmuel felt that the
verse in Mishlei did not apply in his circumstance, he could have
explained to his disciple why this is so.

Here is another example:

A talmid sees his rebbe speak during the repetition of the shemoneh
esrei. It is correct for the talmid to ask his rebbe: “Didn’t we learn
that one may not talk during the chazaras hashatz?” Framing the rebuke
as a question is milder than saying to his rebbe directly: “It is
forbidden to talk during chazaras hashatz.”

As we noted above, someone who sees a person talking during chazaras
hashatz is required to feel tremendous love for this person, so much
so that it pains him to realize that the talker will be punished for
his misdeed. Then, the mochiach tries to figure out what will be the
most effective way of communicating both these feelings and the
message to the wrongdoer.

Brochos 22a:

מעשה ברבי יהודה שראה קרי והיה מהלך על גב הנהר אמרו לו תלמידיו רבינו שנה לנו פרק אחד בהלכות דרך ארץ ירד וטבל ושנה להם.
אמרו לו לא כך למדתנו רבינו

They said to him: Did you not teach us otherwise, our teacher...

So id the Halacha like Aruch Hashulchan, for example:

ראה רבו עובר על דברי תורה – לא ימחה בידו בלשון גנאי, אלא אומר לו: "למדתנו רבינו כך וכך". ואפילו רצה לעבור על איסור דרבנן – מחוייב למחות בידו בלשון הזה. והרואה רבו עושה מעשה, ויש לו להקשות על זה אם הוא איסור דאורייתא – יקשה לו מקודם, ולא יניחנו לעשות המעשה. ואם הוא איסור דרבנן – יניחנו לעשות המעשה ואחר כך יקשה לו, הואיל שאינו יודע בוודאי שעובר אלא שיש לו קושיא – הוה כספיקא דרבנן, ולא נדחה כבוד הרב מפני זה