It was not uncommon for a letter writer to make a personal copy of a letter before sending (see here)—in the first century there was no Google  to maintain a sent folder. Reasons this would be helpful include:
They could resend if the original was lost
They could reference it in reading a response
They could quote it later if they liked the way they said something (I’m actually doing that right here right now)
They could call out a forgery sent in their name
The letters of Cicero occasionally reference the practice of the sender making a personal copy of a letter before sending it.
There are several good reasons to believe Paul had a "sent" folder containing at least some of his letters. These arguments would be less applicable to those Biblical texts that did not originate as letters.
Evidence of a "sent folder" from Paul's letters
Paul’s letters—from as early as the manuscript evidence can be traced—always circulated as a set
The manuscript evidence shows that Paul’s letters tended to circulate as a group. Important examples include P46, Codex Boernerianus, Codex Augiensis, and Codex Claromontanus (see here)
The incredible consistency of the contents of the Pauline corpus (there are 13 letters that are *always there, almost always in the same order) suggests to me that these letters were compiled very early, and the only serious attempt to later add to the set (Hebrews) was never entirely successful because the set of 13 was already known, established, and distributed (fun aside—could this be the origin of the codex? Not sure but it sounds interesting). The significant and increasing influence of the church in Rome over the next few decades would see to it that this set of documents obtained a wide distribution in the empire.
Jack Finegan pointed out that the tremendous familiarity of the Apostolic Fathers with Paul’s letters suggests they must have been circulating as a group within the first century (see here). Based on the fact that there weren’t multiple competing sets, one with say 5 of Paul’s letters, another with 3, and so on, I conclude that we should push the date forward a few decades further still. Some of these churches were relatively close to each other. Why didn’t they create mini-sets of Paul’s letters? Apparently somebody with the whole surviving set pre-empted them.
*Note on P46, the earliest surviving manuscript of a set of Paul’s letters. The outer few leaves of the manuscript have been lost (see here pp. 15-16). Some have suggested the Pastorals (1 Tim, 2 Tim, Titus) weren’t part of this set. The fact that the scribe starts writing more and more cramped and concisely as the manuscript progresses shows that he realized his mistake in not leaving himself enough space and was trying to compensate. If you take out the Pastorals he had more than enough space. So it appears he did intend to include the Pastorals.
The titles on Paul’s epistles are too uniform to reasonably attribute to coincidence
The 13 epistles which begin with the word "Paul" are remarkable in that their titles are incredibly uniform.
If James sends a letter, what do you call it? The epistle of James.
If Peter sends 2 letters, what do you call them? 1 & 2 Peter.
Same basic idea for John. And Jude.
So where are 1st and 2nd Paul??
The letters attributed to Paul are the only letters in the New Testament (and among the very few in all early Christian documents) that are identified by the audience rather than the author.
If you traveled to Thessalonica in the late 50s and asked the local Christian congregation for 1st and 2nd Thessalonians they’d probably be puzzled for a moment, and then take you to visit the graves of the city founders at the local cemetery. If you asked them for the 1st & 2nd letters of Paul they’d know what you were talking about. If the churches called these letters by the name of the author (it would be rather silly if they did not), how do we explain the incredible consistency of the titles referring to the recipients?
My view is that somebody (we don’t know who but Luke & Timothy are likely suspects) compiled Paul’s papers after his death and distributed them around Rome as a set. This set consisted of 13 letters (not Hebrews), and they were labeled by the church/person to whom they were addressed. (David Trobisch & E. Randolph Richards have written extensively on these matters; see a particularly helpful piece by Trobisch here). If this didn’t happen, then we should expect multiple smaller, distinct sets of Paul’s letters floating around rather than one uniform set, as discussed above.
The other major early Christian example of letters titled by audience rather than author would be the writings of Ignatius (circa 107)—and the reason his letters are titled this way is explained in Polycarp’s epistle to the Philippians chapter 13—the were collected as a set right around the time of Ignatius’ death and then distributed as a set.
My argument then suggests the solution to our questions about Paul’s letters is the same as the solution to the letters of Ignatius. These two sets of letters are titled based upon the recipient, not the author, because from very early on they were distributed as a set.
Some of Paul’s letters are missing
The principal competing hypothesis to the view that Paul’s letters were compiled from a “sent folder” is that someone traveled around the empire seeking out Paul’s letters and making copies of them. Not only is this a far less parsimonious solution (and such a trip would be long, dangerous, and expensive), but it suffers from a fatal flaw: where are the missing letters to the Corinthians?
If the traveling letter-gatherer stopped in Corinth (could you really make a serious attempt at collecting Paul’s letters without stopping in Corinth?) why did they only get two of the letters? See this post on this site discussing how many letters Paul wrote to the Corinthians. To make the traveling letter-gatherer hypothesis work we would probably have to assume that the Corinthian church lost 1 or 2 of Paul’s precious letters. While not impossible, I find this difficult to believe.
How does the “sent folder” hypothesis handle the missing letters? There’s no reason to claim that every letter sent by Paul had a copy made and filed among his papers; only that this was a customary practice. The letters that made it into the sent folder survived by securing a place in the original Roman set of 13 Pauline letters, the other letters were lost.
It was not uncommon for a letter writer to keep a personal copy of his correspondence. The evidence from Paul's letters suggests that this was a customary practice by Paul himself.