Unless there is a choir, they would have to. At many of the Eastern Catholic Church Liturgies I have been too, especially among the Ruthenians, there is no choir, and the congregation was able to sing all the responses. In fact, there were two full years where I went daily on weekdays to Mass at an Eastern Church - Ruthenian in Pittsburgh and then later Ukranian in Wilmington, and being generally the only one in attendance on a weekday, made up the choir and congregation. Boy did I learn Slavonic chant quickly - both in Slavonic and English translation!
What is probably most common, at least in America is a mix of choir, solo chanters, and congregation.
As well it should be if possible. In the Roman Rite, the choir (St. Pius X calls them, "The Choir of Levites" to remind us it is really a clerical function) is to sing the proper Antiphons, and the lead the laity in singing the Ordinary.
The key to having the congregation sing along on the responses (and we have a lot of them), is that the fixed portions of the Orthodox Liturgy never change in text (that's just how our liturgies work) -- and wise chanters/choir directors stick to simple chant melodies and don't change them. That way, over time, those who regularly attend learn text and melodies alike by heart. I am personally of the opinion that having books for the congregation actually interferes with this process of memorization.
The Roman Mass is the same. The ordinary is fixed and doesn't change (other than the Creed and Gloria not being sung on weekdays except feasts), and the tones for responses are simple and unchanging. The tones for the Kyrie, Gloria, Credo, Sanctus, and Agnus Dei are around 16 in number (although not all are used very often, some being proper only to Masses of the dead, or of the Blessed Virgin). There is a series of tones for Sundays, i.e. 1 for Easter, 1 for Sunday's after Pentecost, 1 for Advent, 1 for Christmastide, 1 for Lent, etc., and these one quickly learns from repition.
The books are only helpful when one is first learning to follow along. Otherwise, they should only be used for the following the readings and proper (changing) prayers.
For the variable material, this is either sung by the choir, or by a solo chanter. This material is best prayed by the congregation via active and attentive listening. Over time, even some of the variable material is learned by heart (especially troparia).
The learning of this material would be near impossible in the Roman Rite, where every Mass has its own individual chants. Only the most memorable parts, such as the Sequences (Victimae Paschali, Laud Sion, Stabat Mater, etc.), or a handful of commonly used antiphonal chants (such as Gaudeamus Omnes on many feasts of Our Lady), lend themselves to ready memorization.
Ideal liturgies involve a balance of priest(s), deacon (if you're lucky enough to have one), tonsured reader, choir, solo chanter(s), and congregation. It is very natural and fluid.
I don't disagree. For us, Priest, Deacon, Subdeacaon, Lector (for Masses with more than two readings), Cantor(s), Acolytes, Congregation.
No, they wouldn't -- a reader or solo chanter can do the responses in the absence of a full choir. In the Byzantine tradition, this became fairly common during the Ottoman era, since the training of full choirs was very difficult. The association of Byzantine chant with solo psalti grew out of this, but the tradition of Byzantine chant sung by choirs (very beautiful, and by necessity, very un-ornamented) is making a strong come-back.
As to the memorization of variable material, this really only occurs for the troparia and kontakia of the Octoechos on Sundays (they repeat every 8 weeks), and for troparia and kontakia of festal seasons, since these are repeated many times during the services of the feast and during the post-festal season. Some of the Resurrectional stichera for Vespers from the Octoechos becomes memorized by those who attend Vespers regularly, but this is less common.
On a given day, between Vespers, Matins, and Liturgy, there are usually 4 - 8 folio-sized pages of closely spaced text that contain the variable material. This obviously would be very difficult to memorize, even for chanters who were in a monastic setting.
This, incidentally is why Eastern-tradition parishes that claim that the congregation sings "everything" are usually not telling the whole story. In most parishes where this is supposedly the case, the real story is that only Liturgy is being served, and not the other services -- and/or large amounts of variable material are being omitted. It is virtually impossible to do the services with completeness without much of the material being done either by a choir, solo chanters, or both.
In any event, God willing, traditional Western chant will make a comeback in the Catholic church. Most of the Gregorian chant recordings that I have are very sterile and academic -- hardly chants that people could learn and memorize, and it seems that the continuous living tradition of Western chant has been largely broken. Are you aware of recordings that have the kind of simpler Gregorian chant that congregations could sing? And what about in English? (And here's the "hard to please" part -- good English?...)