We begin reciting "Mashib Ha’ru’ah U’morid Ha’geshem" in the Amida in place of "Morid Ha’tal" during Musaf on Shemini Aseret. A number of Poskim addressed the question as to whether one who mistakenly recited "Mashib Ha’ru’ah" during Arbit on the night of Shemini Aseret, or during Shaharit on Shemini Aseret morning, must repeat the Amida. (This question is discussed by the Kaf Ha’haim, Orah Haim 114:7; and by the Erech Ha’shulhan.) As for the final Halacha, Hacham David Yosef, in Halacha Berura, and Hacham Yishak Yosef, in Yalkut Yosef, rule that if one realized his mistake before concluding that Beracha of the Amida (with the blessing of "Mehayeh Ha’metim"), then he should go back and correct his mistake by reciting "Morid Ha’tal." But if he realized his mistake only after concluding that Beracha, then he does not repeat the Amida.

The Shulhan Aruch rules that the congregation does not make the transition from "Morid Ha’tal" to "Mashib Ha’ru’ah" unless an announcement is made in the synagogue to begin reciting "Mashib Ha’ru’ah." This is why it is customary to make a special declaration and conduct a ceremony of sorts before Musaf on Shemini Aseret, announcing the transition from "Morid Ha’tal" to "Mashib Ha’ru’ah."

If one prays at home on Shemini Aseret, such as in the case of an ill patient who cannot come to the synagogue, he should not make the transition to "Mashib Ha’ru’ah" until the time the announcement is made in the synagogue. Hacham David Yosef writes that if there is a Minyan in the neighborhood that prays early, at sunrise, then the ill patient praying at home may recite Musaf with "Mashib Ha’ru’ach" at the time the announcement is made in that Minyan. Even though this person does not normally pray at that Minyan, and prays instead at a Minyan which begins later, nevertheless, he may recite "Mashib Ha’ru’ah" once the announcement is made at the early Minyan.

If, for whatever reason, the announcement was not made in the synagogue before Musaf, then "Mashib Ha’ru’ah" is not recited during the silent Amida. In such a case, the Mishna Berura writes, the Hazan recites "Mashib Ha’ru’ah" in his repetition of the Amida, and this public recitation of "Mashib Ha’ru’ah" takes the place of an announcement. Hacham David Yosef adds that if possible, during the recitation of the silent Amida, the Hazan should recite the first blessing of the Amida quickly and then recite "Mashib Ha’ru’ah" aloud. This would suffice as an announcement, such that the congregation may then recite "Mashib Ha’ru’ah" in their silent prayer.

Summary: On Shemini Aseret, an announcement is made in the synagogue before Musaf to transition from "Morid Ha’tal" to "Mashib Ha’ru’ah U’morid Ha’geshem." The congregation then recites "Mashib Ha’ru’ah" in the silent Amida of Musaf. If one mistakenly recited "Mashib Ha’ru’ah" during the Arbit prayer the previous night, or during Shaharit that morning, he does not repeat the Amida. One who prays at home may recite Musaf with "Mashib Ha’ru’ah" from the time when the announcement is made in the earliest Minyan in the town. If the announcement was forgotten before Musaf, then if possible, the Hazan should recite the first blessing of his silent Amida quickly and then recite "Mashib Ha’ru’ah" aloud, and this suffices as an announcement. If not, then the congregation recites "Morid Ha’tal," and the Hazan recites "Mashib Ha’ru’ah" during the repetition of the Amida, which serves as the public announcement.