From Rosh Hashanah to Yom Kippur, four passages are added to Shemoneh Esrei.
In many congregations, during the Chazzan's repetition, each passage is first recited by the congregation, then the Chazzan. However, on Rosh Hashanah and Yom Kippur, only the last two (Ukhsov and B'sefer) are recited by the congregation as well; the first two (Zachreinu and Mi khamokha) are recited just by the Chazzan. Why?
You are talking about the Chabad custom (and I'm sure others do so as well). The rebbe was asked this question and the answer is printed in Igros Kodesh volume 3 page 5. I will re-post it here with translation:
במכ׳ מכ״ד אלול, במענת על שאלתו לעניו אמירת הקהל זכרנו כו׳ ובספר בעת חזרת הש"ץ. מנהגו מאושר ע"י כ׳׳ק מו"ח אדמו"ר שליט"א בפירוש שהקהל חוזרים וכתוב וחתום ובספר וגם בשויו׳׳ט. ומש׳׳כ שיש חוזרים ג״כ זכרנו ומי כמוך ונ״ל שכן צריך להיות כי מה טעם לחלק בינם ובין וכתוב. כך שמעתי ע"ד מנהגו זה ומעולם תמיהני על חזרת מי כמוך כי הלא אין זה בקשת רחמים כ"א ספור שבחו של מקום. ומה שאין מנהגנו לחזור זכרנו נ"ל הטעם כי הרי בקשת זכרנו לחיים היא בקשת דבר מועט ולכן אין אומרים זה הקהל לאחר שכבר אמרו וכתוב לחיים טובים משא"כ הש"ץ שאמר וכתוב כו׳ כדי שתהא תפלתו שגורה בפיו. ... ואסיים בברכת חוגמח״ט
With regard to your letter of 24 Elul, in response to your question with regard to the congregation’s reading of the phrases zochreinu... u’besefer...when the sheliach tzibbur repeats [the Shemoneh Esreh].
Our custom, as verified explicitly by my revered father-in-law, the Rebbe Shlita, is that the congregation recite u’chsov... and u’chsom... and u’besefer on Shabbos and festivals as well.
You write that there are those [who follow the custom that the congregation] also repeat zochreinu... and mi chamocha...and that it appears to you that this practice should be followed, for what reason is there to differentiate between [these refrains] and u’chsov...?
I have heard about this custom and I have always wondered why the phrase mi chamocha... is repeated, for this is not a request for mercy, but instead a recounting of G‑d’s praise.
The rationale for our custom not to repeat zochreinu...is that the request “Remember us for life” is a lesser request. Therefore the congregation does not repeat it after they have already asked: “Write down... for a good life.” The sheliach tzibbur, by contrast, [recites zochreinu despite the fact that he has said u’chsov, because he recites] u’chsov [only] in order to prepare for the recitation of his [public] prayers. ...
I conclude with blessings for a chasimah and a g’mar chasimah tovah,
