NT scholars typically think "Jesus traditions" were initially transmitted orally. More liberal scholars think this was creative oral tradition; more conservative scholars think this was oral history, based on retentive living memory. Oral cultures foster a retentive memory.
Occasionally, you have a maverick scholar like Alan Millard who thinks writing in the time of Jesus has been neglected. Millard has done original research on the subject, sifting primary sources regarding 1C literacy–especially in Jewish circles. And I think that's a very good angle to take.
I myself espouse the plenary verbal inspiration of Scripture. In addition, I think God enhanced the memories of the disciples (cf. Jn 14:26).
However, let's consider oral history. It's a truism that we remember events better than words. But how accurately do we remember words? Let's take a comparison:
By 1732 he [Jonathan Swift] was noticing a serious deficit in short-term memory: "I often forget what I did yesterday, or what passed half an hour ago." It was a condition he had long foreseen. As early as 1720, when he was walking with Edward Young, secretary to the lord lieutenant at the time, he made a remark that Young put in print much later: "As I and others were talking with him an evening's walk, about a mile out of Dublin, he stopped short; we passed on; but perceiving that he did not follow us, I went back, and found him fixed as a statue, and earnestly gazing upward at a noble elm, which in its uppermost branches was much withered and decayed. Pointing at it, he said, 'I shall be like that tree, I shall die at the top.'"
No reason has ever been given to doubt Young's anecdote; he was a highly principled clergyman as well as a moralizing poet.
…there is corroboration in an independent anecdote from Swift's friend Faulkner: "One time, in a journey from Drogheda to Navan, he rode before his company, made a sudden stop, dismounted his horse, fell on his knees, lifted up his hands, and prayed in the most devout manner. When his friends came up, he desired and insisted on their alighting, which they did, and asked him the meaning. 'Gentlemen,' said he, 'pray join your hearts in fervent prayers with mine, that I may never be like this oak tree, which is decayed and withered at the top, whilst all the other parts are sound.'" Leo Damrosch, Jonathan Swift: His Life and His World (Yale University Press 2013), 460.
Gospel harmonists are sometimes unsure whether similar passages in the Gospels are variations on the same event or similar events. Here we have the same imagery and sentiment, but the setting is different. The wording is quite similar in each case: the difference is that, in the first case, the "withered and decayed" phrase is used by the narrator, while in the second case, it is attributed to Swift. Is that just coincidental? Or did Young misremember that Swift used that phrase? Or did Young remember, but put those words in the mouth of the narrator to introduce the scene?
In any case, we have two independent accounts that convey the same idea, using the same imagery and many of the same words. It's not just the "gist" of what he said. Both accounts preserve some of the very same wording. It's just that in Young's account, some of what is a direct quote in Faulkner (attributed to Swift) is reassigned to the narrator. Young may well be exercising a bit of editorial license, by describing the scene in Swift's words–or perhaps Swift's statement influenced how Young himself remembered the scene.
Whatever the explanation, we're dealing with uninspired recollection of a one-time event, yet in comparing the two accounts, the recollection is both substantively and verbally accurate.