Interestingly, if one does not accept that Luke’s Gospel is using Matthew as a source, and thus accepts the two-source hypothesis first proposed by Heinrich-Julius Holtzmann, along with the hypothesis that two other sources in addition to Mark and Q were used by Matthew and/or Luke, along with agreeing that both Matthew and Luke use a different source for their birth-narratives, then that seems to imply that Matthew and Luke both used two different birth-narrative sources, let us call them M1 and L1, each of which were independent of each other. However, that means that M1 and L1 both independently attest to the virgin birth.
Somebody may want to say that perhaps M1 and L1 were both dependent on at least one source in common, call it Q1, which attested to the virgin birth, but aside from being convoluted, there is also no good reason to suppose that M1 and L1 are not entirely independent. After all, whatever source Matthew used for his infancy-narrative, it apparently had nothing in common with whatever source Luke used for his infancy-narrative, the virgin birth notwithstanding. Moreover, at least we have some reason to think that Matthew and Luke were working with sources (for example, material common to both of them, not found in Mark, is at least evidence of at least one other source – though I think it could be adequately explained more parsimoniously by arguing that Luke was dependent upon Matthew).
Now, perhaps somebody could argue that the virgin birth became a popular element of earliest Christian tradition, and thus just happened to find its way into at least two different and unrelated/independent accounts. However, given that conservative scholarship situates the writing of Matthew as early as the late 50’s or into the 60’s, and that more liberal scholarship situates the writing of Matthew no later than the 80’s, and that conservative scholarship situates the writing of Luke as early as 61-62 A.D. (with what I consider to be relatively strong arguments), and liberal scholarship situates the writing of Luke no later than the 80’s – that means that at very best the tradition of the virgin birth was circulating for about 50 years (assuming traditions about Jesus – or at least his infancy – began no earlier than after the time of his death) before finding it’s way in Matthew and Luke. Moreover, it must have found it’s way into M1 and L1 earlier than that. However, some have argued that if the story of the virgin birth were a creation, then it must have come from the Hellenistic world, because it was not part of the Jewish paradigm. I don’t accept that argument, but supposing I did (and at least the story plausibly wouldn’t have become popular in a Church which was for the most part Jewish instead of for the most part Greek et al.), it would make it more likely that the story could only have been a creation of the Church after Nero and the disassociation of the synagogue and temple-worship community of observant Jews, and those Jews who professed Jesus as the Messiah. This, however, was no earlier than 64 A.D. That narrows the time in which the story could have been created to a period of less than 16 years. One may be stuck arguing, then, that the story of the virgin birth arose and gained popularity, found it’s way into M1, along with L1, and then by extension into Matthew and then Luke, all within less than 16 years. Moreover, if it was so widespread, it is a wonder that it doesn’t appear in any other infancy narratives (infancy Gospel of Thomas, Proto-evangelium of James, Gospel of Hebrews, etc. [to the best of my knowledge, but I should double-check – for the sake of argument let’s assume I’m right]*).
That’s pretty uncomfortable. Perhaps then one could abandon the popular solution to the synoptic problem and just claim that Luke is dependent on Matthew (or, I suppose, that Matthew is dependent on Luke). Somebody may argue in vain that L1 and M1 must not be independent, such that either L1 is dependent on M1, or vice versa – however, no similarity other than the virgin birth would exist between them as far as textual criticism can/could determine. It seems almost inevitable then that if one does not accept that Luke was dependent on Matthew, or Matthew on Luke, and that both Matthew and Luke used independent infancy-narratives (M1 and L1), then M1 and L1 both attested to the virgin birth independently.
However, the best explanation for multiple sources independently attesting to Jesus’ virgin birth may just be that the story was primitive (arose before the 60’s). However, if this is so, then it begins to look very likely that the virgin birth was a historical event, since it seems as alien to a Jewish mindset as the death and resurrection of the Messiah.
Of course, we could spin however many theories to get us out of this mess (maybe Matthew was redacted by the Lukan community, or vice-versa), but it seems at this point that the liberal is just wiggling to escape the implication of the miraculous.
*Note: I do think an argument can be made that John 1:13 can be read, according to a minority variant in the singular, as a reference to the virgin birth. In any case, there may be other reasons to believe that the Johannine community behind the Gospel of John was aware of (and perhaps embarrassed at) the virgin birth. So, that may be something.