The idea that the gospels initially circulated anonymously is popular in skeptical circles, as illustrated by the recent exchange between Bart Ehrman and Tim McGrew. However, it's an idea that's heavily contradicted by the evidence. Even moderates and liberals have been voicing doubts about it. As I documented in an earlier post, non-Christian, heretical, and Christian sources were naming the gospel authors or describing them with terms like "apostle" and "disciple" (in the highest sense) before Irenaeus commented on the subject in Against Heresies. Those sources go back as early as the first century.
What I want to do in this post is use Polycarp as an example of somebody who probably named the gospel authors (and other New Testament authors) and thereby influenced later authorship attributions, even though he doesn't name the gospel authors in any of his extant material. In other words, we have good reason to think the authorship of the gospels and other New Testament documents was widely discussed prior to Irenaeus, even among sources who don't name or describe the authors in our extant literature. Polycarp is an example.
Even critics like Ehrman acknowledge that the gospels and other New Testament documents are widely attributed to their traditional authors from the late second century onward. The sources making those authorship attributions are highly diverse in their locations, theologies, backgrounds, personalities, etc. The question is, what preceded that widespread agreement? If individuals like Polycarp were saying that the gospels, for example, were anonymous or were attributing them to authors other than the traditional ones, why would the traditional attributions be so popular just afterward, with no trace in the historical record of people as influential as Polycarp having advocated a different view? The best argument people like Ehrman could make to explain the evidence would be that men like Polycarp didn't comment on authorship issues much or at all. That would explain how they could have not held the traditional view, yet didn't leave much trace of a non-traditional view in the historical record. But how plausible is that kind of scenario?
Polycarp apparently was a disciple of the apostle John and had met at least one other apostle. For a discussion of the evidence to that effect, see here. He traveled widely and was often in contact with Christians outside of his city, judging by other sources' accounts of his life and his comments in his Letter To The Philippians. See, for example, his references to travel in his letter (13) and the detailed knowledge he shows of the history of the Philippian church and other churches (11-14). So, he would have been in a good position to reliably comment on authorship issues.
And he does so explicitly in some cases. In his Letter To The Philippians, he refers to a letter Paul wrote to the Philippian Christians (3) and refers to Ephesians as scripture (12). Given the high view of scripture held by the early Christians and their tendency to accept authorship attributions made within a scriptural document, Polycarp's acceptance of Ephesians as scripture implies that he accepted the Pauline authorship of the document.
How familiar was he with the other documents of the New Testament? Bruce Metzger, in a work Ehrman refers to as "the standard authoritative scholarly account" of the New Testament canon (Misquoting Jesus [San Francisco, California: HarperSanFrancisco, 2005], n. 10 on 220), concludes that Polycarp shows familiarity with at least eleven New Testament documents in his Letter To The Philippians alone (The Canon Of The New Testament [New York: Oxford University Press, 1997], 62). Metzger refers to how Polycarp's mind was "saturated" with ideas and phrases from the New Testament (63). So, it seems that Polycarp was highly familiar with at least a large percentage of the documents and often discussed them with other individuals and groups (as illustrated by his exchanges with Ignatius and the Philippian church).
Irenaeus, a disciple of Polycarp, gives us a lot of significant information about Polycarp's life. Not only had he met Polycarp and been taught by him, but he also refers to multiple extant letters of Polycarp (Fragments, 2), meaning that he had access to some literature from Polycarp that we don't have today. Irenaeus tells us:
"I can even describe the place where the blessed Polycarp used to sit and discourse— his going out, too, and his coming in— his general mode of life and personal appearance, together with the discourses which he delivered to the people; also how he would speak of his familiar intercourse with John, and with the rest of those who had seen the Lord; and how he would call their words to remembrance. Whatsoever things he had heard from them respecting the Lord, both with regard to His miracles and His teaching, Polycarp having thus received information from the eye-witnesses of the Word of life, would recount them all in harmony with the Scriptures." (ibid.)
Irenaeus is referring to the regular practices of Polycarp, not just things Polycarp did on one or two occasions. It seems that Polycarp often discussed issues highly relevant to the gospels (Jesus' miracles and teaching) and did so consistently with the scriptures. If hearers of Polycarp, like Irenaeus, were comparing Polycarp's comments to the scriptures (which included the gospels for Irenaeus), it's unlikely that Polycarp would have been unaware of that context or have not commented on it. It would be hard to believe that he didn't frequently make reference to the authorship of the gospels in such contexts.
Regarding a dispute over the celebration of Easter, Irenaeus wrote:
"And when the blessed Polycarp was sojourning in Rome in the time of Anicetus [a Roman bishop], although a slight controversy had arisen among them as to certain other points, they were at once well inclined towards each other with regard to the matter in hand, not willing that any quarrel should arise between them upon this head. For neither could Anicetus persuade Polycarp to forego the observance in his own way, inasmuch as these things had been always so observed by John the disciple of our Lord, and by other apostles with whom he had been conversant; nor, on the other hand, could Polycarp succeed in persuading Anicetus to keep the observance in his way, for he maintained that he was bound to adhere to the usage of the presbyters who preceded him. And in this state of affairs they held fellowship with each other; and Anicetus conceded to Polycarp in the Church the celebration of the Eucharist, by way of showing him respect; so that they parted in peace one from the other, maintaining peace with the whole Church, both those who did observe this custom and those who did not." (Fragments, 3)
Notice the further evidence that Polycarp traveled (he was in Rome for a while), was in contact with other individuals and churches, was involved in many disputes (note Irenaeus' comment about how "slight controversy had arisen among them as to certain other points"), etc. The Easter dispute in question was one that involved the gospels to some extent, such as with regard to issues surrounding the timing of Jesus' death and resurrection.
Here's something Irenaeus writes elsewhere about Polycarp's time in Rome:
"He it was who, coming to Rome in the time of Anicetus caused many to turn away from the aforesaid heretics to the Church of God, proclaiming that he had received this one and sole truth from the apostles—that, namely, which is handed down by the Church." (Against Heresies, 3:3:4)
When Irenaeus describes that truth handed down by the church (ibid., 1:10:1-2), he refers to beliefs like the virgin birth and Jesus' resurrection, which are found in the gospels.
Furthermore, when Polycarp attended eucharistic and other church services in Rome and elsewhere, Justin Martyr gives us some idea of what he would have experienced:
"And on the day called Sunday, all who live in cities or in the country gather together to one place, and the memoirs of the apostles [identified as the gospels in the previous section of Justin's work] or the writings of the prophets are read, as long as time permits; then, when the reader has ceased, the president verbally instructs, and exhorts to the imitation of these good things. Then we all rise together and pray, and, as we before said, when our prayer is ended, bread and wine and water are brought, and the president in like manner offers prayers and thanksgivings, according to his ability, and the people assent, saying Amen; and there is a distribution to each, and a participation of that over which thanks have been given, and to those who are absent a portion is sent by the deacons." (First Apology, 67)
Not only would Polycarp hear the gospels being read and discussed during such church services, but notice, also, that the inclusion of multiple gospels in church services would have required a means of distinguishing among them. The earliest sources we have, including the earliest relevant gospel manuscripts, tell us that the gospels were distinguished by means of author names ("The Gospel According To Matthew", "The Gospel According To Mark", etc.). And notice that the gospels weren't just read during church services, but also were the objects of instruction that followed the reading. In both contexts, the names of the gospels' authors would have come up. Again, it would be difficult to believe that Polycarp never discussed authorship issues in such contexts, never corrected false authorship attributions that he heard, etc.
Here's another aspect of Polycarp's life recounted by Irenaeus:
"And Polycarp himself replied to Marcion, who met him on one occasion, and said, 'Do you know me?' 'I do know you, the first-born of Satan.'" (Against Heresies, 3:3:4)
Marcion was known largely for holding to an edited version of Luke's gospel while rejecting the other three gospels. How would Polycarp become familiar with Marcion and his beliefs and practices enough to recognize him by his physical appearance and respond to him as described above without, in the process, giving much thought to or commenting much on the authorship of the gospels (and other relevant documents, like Paul's letters)?
Irenaeus also tells us that Polycarp was a companion of Papias (Against Heresies, 5:33:4). Since Papias shows familiarity with multiple gospels and their authorship and comments on those subjects prominently (Richard Bauckham, Jesus And The Eyewitnesses [Grand Rapids, Michigan: Eerdmans, 2006], 12-38, 202-39, 412-37), how likely is it that his companion, Polycarp, was ignorant of all of those issues or rarely or never commented on them? Furthermore, Papias cites a man he refers to as "the elder" as a source of information on the authorship of the second gospel, and that man probably was the apostle John. Since Polycarp probably was a disciple of John, his receiving the same information from John makes more sense than his having received no or different information.
In summary, we see a large number and variety of circumstances in which Polycarp probably would have interacted with other people on authorship issues, including the authorship of the gospels. In some cases, like the disputes surrounding Easter, the reading of and commenting upon the gospels in church services, and the disputes surrounding Marcion, it's highly probable that gospel authorship would have come up frequently. If Polycarp wasn't silent about authorship issues, and the people he influenced affirmed the traditional authorship attributions while showing so little trace of alternative views, the best explanation is that Polycarp affirmed those traditional attributions.