OK I went and researched myself and reached some conclusions.
In order to answer this question, we first have to ask whether back then there was some organized doctrine of musical theory and whether there was some sort of pedagogical tradition.
The French Benedictine monk Hucbald was the first to write a theory book. At the end of the 9th century he published an essay called "On the Principles of Music", where he redefined the Greek Modes and thus created a theoretic-harmonic literary source on which a European composer could lean. We must notice that by "harmonic" we don't mean common western cadential chord progressions, but just choosing the modal framework in which to work. Chord progressions are not relevant (yet).
It is well worth mentioning that at this point we are some 100 years before Guido d'Arezzo brought the gospel of modern notation so hucbald's musical language tools are quite limited. Nevertheless, the way we today define modes isn't far at all from Hucbald's and the principal of tonality (as in having a tonic and hierarchy between notes) is already there! However, the Ionian (major) and Aeolian (minor) modes aren't there and we'll have to wait 650 years for them to appear. Anyway, the institutionalizing of the modal approach became more and more sophisticated as time went on.
Concerning the existence of some pedagogical tradition, today we know that in Medieval Times, choir boys used to have ear and voice training lessons in churches and academies, but was there any tutoring for the chant composers? There is no uniform answer for Medieval Times. In general, we can put the line elegantly where music went through secularization and polyphonization - in research we find the term "music teacher" only from the 12th century onwards (I mean those who teach subjects other than notation, ear and voice training). This is not surprising, the moment music stopped being solely in service of words (as is the case with monophonic liturgical chants), it became more complex and much greater skill was required in order to compose. As a result the need for teachers arose.
The first (obvious and quite well known) conclusion is that Medieval Times aren't uniform in their musical and historical-musical characteristics. How can we generalize over more than an inadequately documented thousand years of advancements and innovations? Up until the 12th century, the identity of composers of tunes and chants isn't known to us at all, we have a good guess at best...
Now, did Hildegard von bingen (had monophonic chant style and operated from beginning to middle of 12th century) study musical theory? According to the literature apparently she studied playing and reading music from a noblewoman named Jutta but concerning composing - she was an autodidact. It is a reasonable assumption that a very very basic theoretical knowledge (the notion of mode), a talent for composition and attendance in places where they played European music (churches and other holy places) was enough for composing well in monophonic style.
And now, did the great medieval composers which came afterwards (post secularization and polyphonization), such as Machaut, Landini, Dunstable and Dufay, have some kind of theoretical training? It is likely they did. With a rising complexity inherent to pieces composed at the time, naturally the need for knowledgeable and professional tutoring arose. Teachers HAD to hold musical-theoretical knowledge. I didn't mention Leonin and Perotin (they were between Hildegard and Machaut). They were pioneers in the polyphonic craftsmanship so there is no point in discussing who taught them what a mode is and how to write in polyphonic texture.
At this point we can start and try to answer the initial question: In that era, what was the composers' level of awareness of musical theory and the composition techniques underlying their music. First of all, every job requires a work method - from a construction worker to an academic scholar. Work method is at least order of operations and usually means rules. Thus, we can determine that with more musical complexity, the work method became richer and obviously composers who had to master a wider theory, were more aware of the musical theory underlying their music and had composition techniques at their disposal. At the end, this claim is quite soft. I'm just claiming that Hildegard von bingen probably didn't study composition and Machaut probably did (to some extent at least).
-The Oxford History of Western Music part 1 chap. 3, R. Taruskin
-Craig Wright (1986). Leoninus, Poet and Musician. Journal of the American Musicological Society.
-White, John (1998). The Musical World of Hildegard of Bingen. College Music Symposium Vol. 38