In this week's Parsha, the Torah commands us to count seven complete weeks from the day on which the Korban Omer is brought (ויקרא כ"ג:ט"ו). In that Posuk, the Torah uses the language "וספרתם לכם...," "and you shall count for yourselves..." from which the Gemara in Menachos (.דף ס"ה) derives that every person is commanded individually to participate in this Sefirah, this counting. Indeed, the Rambam (פרק ז' מהל' תמידין ומוספין הלכה כ"ד), the Tur (אורח חיים סימן תפ"ט), and the Shulchan Aruch (שם סעיף א'), all record this statement as the Halacha, ruling that each person must count Sefirah for himself. With this in mind, a question arises as to whether one can fulfill the Mitzvah of Sefirah through the mechanism of שומע כעונה, which means that one can be credited with having said something simply by hearing another person say it, as long as both the person doing saying the words and the person hearing the words have the intent to fulfill the Mitzvah together. This mechanism works for many Mitzvos, such as Kiddush and Birchas HaMazon, among others; for Sefirah, however, since the Gemara (שם) states that each person must count by himself, perhaps one must indeed do it alone without utilizing the idea of שומע כעונה.
The Chok Yaakov (שם ס"ק ד'), quotes from a Teshuvah of the Rashba (שו"ת הרשב"א חלק א' סימן קכ"ו) that even though שומע כעונה usually is effective, this is only the case with regards to something which is a Beracha. As for the actual Sefirah, however, one can not fulfill the Mitzvah through שומע כעונה, but each individual person must rather count for himself. He proves the validity of this conclusion by comparing this phrase "וספרתם לכם" regarding Sefiras HaOmer to the requirement incumbent upon every individual to take a Lulav on Sukkos based on the Posuk which says "ולקחתם לכם" (שם פסוק מ'); just as the Mitzvah of Lulav must be performed by each person individually, so too must the Mitzvah of Sefiras HaOmer. The Magen Avraham ()שם ס"ק ב' rules accordingly, as does the Be'er Heitev (שם ס"ק ג') who points out, though, that if a person does indeed listen to someone else count the Omer, he fulfills the Mitzvah בדיעבד, meaning in a less than optimal fashion, provided that both the person who counted the Sefirah and the listener had the proper intent. With regards to the Beracha before Sefiras HaOmer, Rabbi Akiva Eiger (הגהות רעק"א שם על המג"א ס"ק ב') actually writes that it is preferable to use the mechanism of שומע כעונה, since by doing so, one will also fulfill the dictum of ברוב עם הדרת מלך, meaning that it is best to glorify the King in public.
The Pri Chodosh (שם ס"ק א') explains that when the Gemara (שם) stated that the Mitzvah of Sefirah is incumbent on every individual, it did not mean that every individual must do the counting himself and that it can not be fulfilled through שומע כעונה by listening to the Sheliach Tzibbur. Rather, the Gemara (שם) made this statement in order to counter a notion that one may have inferred from another Mitzvah which is similar to this. In Parshas BeHar, the Torah uses the phrase "וספרת לך," "and you shall count," with regards to the Mitzvah of counting seven Shemittah cycles until the fiftieth year of Yovel (שם כ"ה:ח'). This Mitzvah of counting to the Yovel is incumbent only on Beis Din; the aforementioned Gemara in Menachos (שם) thus tells us that in this case, regarding Sefiras HaOmer, the Mitzvah applies to all Jews and not only to Beis Din. In fact, the Rambam in his Sefer HaMitzvos (מצות עשה קס"א) makes this distinction, and the Aruch HaShulchan (שם סעיף א') presents it as well. It is for this reason that the Pri Chodosh (שם), despite ruling originally that it is preferable that one should count by himself, concludes that even לכתחילה, to fulfill the Mitzvah optimally, one may use the mechanism of שומע כעונה. The Machatzis HaShekel (שם ס"ק ב') uses the same logic to rule that שומע כעונה does work here, and therefore it's as if the person listening actually counted himself. He also explains that the Rashba, in the Teshuvah cited above (שם), where he rules that one can fulfill only the Beracha of Sefirah through שומע כעונה, could agree that one can indeed fulfill the actual Sefirah through שומע כעונה, but he did not want to make a blanket statement like that for another reason. Since the widespread practice today is that everyone counts Sefirah for himself, the Sheliach Tzibbur might not have the proper intent to recite the Sefirah with anyone else in mind, in which case שומע כעונה would not work. The Rashba therefore did not want to encourage anyone to rely on the Sefirah of the Sheliach Tzibbur. However, if indeed both people do have the proper intent, undoubtedly even the Rashba would rule that listening is sufficient even to fulfill the Mitzvah of counting.
Interestingly, the Pri Megadim (במשבצות זהב שם ס"ק ב') derives from the above Posuk in our Parsha (שם פסוק ט"ו) both the fact that one may not fulfill one's Mitzvah via שומע כעונה, and the point that the Mitzvah of counting is not limited to Beis Din. He writes that from the word "וספרתם," "and you shall count," we learn that the Mitzvah is incumbent upon every Jew and not on Beis Din alone, while from the word לכם"," "for yourselves," we learn that every Jew must do the counting by himself and not through שומע כעונה. The Mishnah Berurah (שם ס"ק ה') cites both those who say that the word וספרתם teaches us that שומע כעונה does not apply, and those who say that it teaches that this Mitzvah belongs to each individual and not Beis Din. In any case, he concludes that it is the Minhag nowadays not to fulfill the Mitzvah through the Sheliach Tzibbur, but that everyone should rather count for himself. However, in his Biur Halacha (שם בד"ה ומצוה) he notes that בדיעבד, if one hears someone else count and has the proper intentions, he should count again, but without a Beracha, since technically he may have fulfilled his obligation.