Does someone who’s not part of a Minyan have to answer the prayer?
The Shulhan Aruch (O.H. 55:20) says that if one is in the vicinity of, but not a part of a Minyan, one may answer Kaddish or Kedusha that is being recited in that Minyan. The Shulhan Aruch seems to imply that one is not obligated to answer, but may do so if one wishes [as to why one would not be able to answer, see below]. Rabbi Ben Zion Abba Shaul (Or Lezion, ch. 45, § 12) says that if one is learning Torah, for example,in an adjoining room or in the sanctuary where the Minyan is praying, one does not have to interrupt one’s learning to answer “Amen”. On the other hand, Rabbi Shlomo Zalman Auerbach (Halichot Shlomo, ch. 9, § 6) says that if one is in another room, one need not interrupt one’s learning to respond to parts of the prayer. If one is in the same room as the Minyan, however, he says that one would not need to respond to Kaddish, but that it would be unseemly to ignore Kedusha, Modim and Berechu and that, therefore, one would need to answer.
Tangentially related to this is the concept of Amen Yetoma, or an orphaned Amen. Amen Yetoma refers to an “Amen” that one recited but did not hear the blessing or part of the prayer, and thus the Amen is “orphaned” from that blessing, as it were. The Rama (O.H. 56:1) states that if one walks into a Minyan and hears the congregants answering Kaddish, one may also respond, even though one did not hear the Kaddish being recited. Since the person knows that it is Kaddish that is being responded to, the Amen is not orphaned. This is similar to the Gemara’s (Sukkah 51b) description of a very large synagogue in Alexandria, Egypt in which flags would be waved in order for the distant congregants to be able to know when to answer the prayer.
In a similar vein, Rabbi Ovadia Yosef (Yehave Da’at, vol. II, § 68) says that if there is a Minyan and one is listening to the Minyan live on the radio, one is able “Amen” to the different parts of the prayer, and it would not be considered an Amen Yetoma. This would seem to apply if one is watching live on television or if it is being broadcast live on the internet. It should be noted that one’s obligation for a particular Mitzvah cannot be fulfilled in this method, such as if one were to hear the blessings and recital of Megilat Esther or Havdala.
Summary: If one is not participating in a Minyan but is in the same area, one is not obligated to respond to Kaddish, but may do so if one wishes. One should, however, respond to Kedusha, Modim and Barechu. One may answer “Amen” to a prayer or blessing that is recited over a live broadcast.