Rosh Hashanah 34a: This Tanna derives the sounds of the shofar from the silver trumpets used in the desert [rather than from Yovel]. “And you shall blow a teruah” (Bamidbar 10:5) – a blow separately and a teruah separately.
Orach Chaim 585:2, Rema: It is better to blow the shofar on the right side of one’s mouth, if possible. Mishnah Berurah: The reason is because it says, “The Satan was standing on his right side to accuse him” (Zechariah 3:1). Biur Halacha: I also heard in the name of the gaon Rabbi Meir Simcha Hakohein a good reason for this, because the Gemara derives the sounds of the shofar from the trumpets used for war, and regarding the war of Gideon it says that they held the torches in their left hands and the shofaros in their right hands (Shoftim 7:20).
ר”ה לד ע”א: והאי תנא מייתי לה בגזירה שוה ממדבר, דתניא: (במדבר י) ותקעתם תרועה ־ תקיעה בפני עצמה ותרועה בפני עצמה.
או”ח תקפ”ה ס”ב: וטוב לתקוע בצד ימין. ביאור הלכה: עיין במ״ב, עוד שמעתי בשם הגאון מהר״ר מאיר שמחה הכהן טעם נכון כי בש״ס ר״ה ד׳ ל״ד ילפי לה לתקיעת שופר מחצוצרות המלחמה ובקרא (שופטים ז) אצל מלחמה בגדעון כתיב ויחזיקו ביד שמאלם בלפידים וביד ימינם השופרות לתקוע וכו׳.
Rabbi Mordechai Elefant, rosh yeshiva of Itri, related: I once visited Rav Shach with a friend of mine ten or twelve years ago [mid-1990s]. I was in a bit of a hurry, but Rav Shach had a story he really wanted to tell me so he held me back. In the Biur Halacha in Hilchos Shofar, the Chofetz Chaim writes that the shofar should be blown from the right side. He writes that he heard a source for this from Reb Meir Simchah. It was a verse in Tanach. Rav Shach made a big to-do about this. He told us the reason the Chofetz Chaim did this even though he didn’t cite other contemporaries. He wanted to make sure that people didn’t think he was angry at Reb Meir Simchah. It was at the time that the Russian government passed a law that Russian had to be part of the curriculum of the Volozhiner Yeshivah. All the gedolim held a big meeting to decide whether they should accept the law and keep the yeshivah open, or not accept it and close the yeshivah down. The Lubavitcher Rebbe of the time attended, too. He was close with all of the gedolim; he was like a Litvak. They decided to close the yeshivah. The Chofetz Chaim addressed the gathering to say that the yeshivah should close. Reb Meir Simchah made a comment to the effect that a little Jew from a little town shouldn’t get involved in an issue like this; he dismissed him.
The Chofetz Chaim was very close with Reb Chaim Soloveitchik. Reb Chaim told him, “Don’t feel bad. Your book, Shmiras Haloshon, is a more important work than his Or Sameach.”
Rav Shach concluded, “That’s the reason I think the Chofetz Chaim cites Reb Meir Simchah. He wanted to show the world he’s not mad at him even though he insulted him.” Rav Shach made a big deal of this point. This is the authentic version of the story. Another book mentions it, but distorts it.
I got home, and I got a call from Hillel Zaks’ oldest brother, Hershel. He asked me to do him some favor with the government. I made the necessary phone calls and then I called him back. He was a grandson of the Chofetz Chaim. I mentioned to him that I had just visited Rav Shach and he told me this interesting story. He said, “My father asked the Chofetz Chaim exactly that question, and the Chofetz Chaim gave exactly the same answer that Rav Shach gave.” I figured that there was no way I could make Rav Shach happier than with that bit of information.
The next day I was at a wedding in Bnei Brak, so I hopped over to Rav Shach. I walked in and there seemed to be nobody in the house besides him. I wanted to set him up a bit, and I started to say I had something to tell him. I wanted to make it really dramatic. He said, “Out with it already.” So I tell him and he jumps up as high as the chandelier and hugs me and kisses me. He said, “You’ve added a piece onto my life.”
Source: Rabbi Mordechai Elefant’s Memoirs, p. 18