The Hesed La’alafim (Rav Eliezer Papo, 1786-1827) writes (listen to audio recording for precise citation) that the Musaf prayer should be treated no differently from other prayers, and the congregation should both recite the Amida silently and listen to the Hazan’s repetition. The author observes that some congregations seek to shorten the prayer service by skipping part of the Musaf prayer. He writes that this was especially common on Shabbatot when Semahot were celebrated, and the congregation sang many songs which took a great deal of time. They would then shorten the Musaf prayer because of the late hour. The Hesed La’alafim strongly condemns this practice, applying to such people the harsh words spoken by the prophet Yeshayahu (43:22), "Ve’lo Oti Karata Yaakob Ki Yagata Bi Yisrael" – that the people’s prayers are not even considered actual prayer to the Almighty. As such, the Hesed La’alafim writes that those who are in a position to object to this practice should do so.
We might also add that those who skip the repetition of the Amida at Musaf forfeit the opportunity to hear Birkat Kohanim, which is a blessing we receive from the Almighty Himself. This is the most significant blessing that anybody can receive. People are willing to travel across the ocean, some ten hours of air travel in each direction, in order to receive blessings from prominent Rabbis in Israel, yet they do not want to spend the four or five minutes to hear the repetition of the Amida, which includes a blessing from G-d Himself. People travel to the Orient – eighteen hours in each direction – for their livelihood, but do not have the patience for the repetition of the Amida when we are blessed, "Yebarechecha Hashem Ve’yishmerecha," which is a blessing for material prosperity. We need to keep our priorities straight, and recognize that listening to the repetition of the Amida – including the Amida of Musaf on Shabbat – is time well-spent, both because of the Halachic requirement to hear the repetition, and because of the inestimable value of Birkat Kohanim.
