It All Depends on How You Define the Word "Count"

The Torah tells us to "count" things several times. Sefiras HaOmer is a famous example of counting. There is also the "7 clean days" counted by a zavah. Finally, there is also the counting of years leading up to Yovel.

The Rambam in Shemittah V'Yovel 10:1 implies that Beis Din HaGadol has a mitzvah to actively count the shemittah years until reaching yovel. The Rambam does not mention making a berachah on this count.

However, Tosafos in Menachos 65b (d"h Usfartem) does mention counting with a beracha. Although Tosafos in Menachos is mesupek in the end, Tosafos in Kesuvos 72a is definitive that Beis Din does make a beracha on the count towards yovel.

Interestingly, Rabbi Akiva Eiger (Shu"t Kamma 29) relates making a beracha to having to count out loud (as opposed to just counting in one's brain). R' Akiva Eiger says that since the Rambam mentions no beracha perhaps the Rambam holds that Beis Din does not have to vocally count the years of shemittah. I'm not sure how a unit of people like Beis Din would count in their brains - maybe they could put up a sign what year it is or something like that. In any case, the pashtus of the Rambam is that Beis Din does count out loud - but we would still have to deal with the beracha issue.

What about the count by zavah? R' Akiva Eiger mentions that the poskim unanimously agree that the zavah doesn't actually need to count the days - she just needs to "keep count" - meaning to keep track of how many clean days she has had. However, it is not totally unanimous.

The Shelah is mentioned in a note on the side of Tosafos in Kesuvos 72a that he understands Tosafos to hold that a zavah actually does count the seven days - she just doesn't make a beracha. The other way to read Tosafos is that Tosafos is saying the Zavah doesn't have to count at all.

The Noda B'Yehuda (Tinyana Yoreh Deah 123) famously comments that the Shelah in his holiness made a mistake and added mitzvos that don't exist - for there is no mitzvah for a zavah to actually count the days. She only needs to keep track.

Besides the Shelah I saw that Footnote 4 on the same R' Akiva Eiger mentions that the Sefer Agudah in the name of the Ri (Menachos 86 Siman 32) also holds that the Zavah has a mitzvah to count. So again, it's not totally unanimous.

Finally Tosafos in Menachos comments on the grammer of the pesukim. By Yovel it says "and you will count" in singular. Therefore, it refers to Beis Din. However, by Sefiras HaOmer it says "and you will count" in plural - which means each individual must count.

However, we should note that the word "count" is mentioned in reference to sefiras haomer more than once and the singular is also used. In fact, the Sifri at the end of Parshas Reah implies that the mitzvah of sefiras haomer is two mitzvos - one on Beis Din and one on the individual. This is consistent with the two usages in the pasuk. However, the Gra changes the girsa of the Sifri that the count of Beis Din is only a hava amina in the Sifri - and not l'maskana. Nevertheless, the Chizkuni at the beginning of Parshas BeHar seems to say even l'maskana that there are two mitzvos by Sefiras HaOmer - one on Beis Din and one on the individual.

*For more on this topic see Sefer Eretz HaTzvi by Rabbi Hershel Schachter Siman 3.