Speaking for example of Pliny's letters (assuming they were typical), how and by whom were the letters first transcribed? I assume that Pliny had someone copy each letter before he sent it? Afterwards, where and how were the letters preserved? Were they published in some way, or was there only one copy of them until they began to be copied in the post-Roman period (by monks, I assume)? Were they in a library? In a box of manuscripts? Is there some way to trace the entire history of a typical letter?
It's easier to look at Cicero, because we have letters, speeches, and other writings of his preserved, and we know so fucking much about his habits of publication – because he wrote about it.
Cicero published his political and judicial speeches, 58 of which survive in tact, and philosophical works during his life. His friend Atticus owned a number of specialized slaves for copying manuscripts for the purpose of putting books into circulation, though Brill's New Pauly says this was not primarily a for-profit activity. Nevertheless Atticus is the most famous publisher from the ancient world. From at least 55 if not earlier Cicero and Atticus produced and circulated edited versions of Cicero's speeches and philosophical dialogues, and Cicero often wrote Atticus giving him updates about the progress of his work. So for instance, while the text of most of the Republic was lost until... 1820? we knew about its content, Cicero's changing plan for the work, the characters, etc., and we know when he changed his mind about these things. It should be noted that it's not clear how closely Cicero's published speeches reflect the speeches he actually gave; there's a famous story about his defense of Milo: due to the political violence following the assassination of Clodius by Milo, the senate suspended elections and appointed Pompey sole consul. Pompey occupied the city with troops as a peace-keeping measure, but Cicero feared Pompey considered him an enemy, and when Pompey's troops showed up at the trial, Cicero supposedly was so terrified he was unable to deliver his defense of Milo at all. Milo lost and went into exile at Marseille. When he saw Cicero's published defense he is supposed to have joked that if Cicero had delivered that speech, he [Milo] would never have had to opportunity to taste Marseille's world-class oysters.
Cicero around 44 had decided to publish his letters with Atticus and began editing them for publication as an anthology. Plato and Aristotle had anthologies of their letters collected and in circulation by the 2nd century BCE. Cornelia's [the mother of the Gracchi] letters were also circulating as an anthology, so Cicero is not unique at all in this. There are some 900 letters, including 100 letters from others to Cicero, divided between four bodies of work; Cicero's Letters to his Friends, the Letters to his Brother, to Brutus, and to Atticus. There must have been copies of at least the letters Cicero sent preserved in his own records, and I suspect we even have something of the organization of his archives preserved. The letters are organized by correspondent, not chronologically. This makes sense; Cicero could consult the letters to a particular person to remind himself of their previous conversations without fishing through all the letters he wrote in the meantime; it's the same thing we do with email threads. The reader might prefer a chronological organization, to see what Cicero wrote and to whom on a day-to-day basis. Cicero was killed before he finished editing these letters, and his slave/secretary Tiro finished the work and published them with Atticus after Cicero's death. That's the Letters to Friends, I mean. I don't know the publication history of the Letters to his Brother or the Letters to Brutus.
The letters to Atticus had a very different publication. These were published in the 1st century CE. They contain no letters from Atticus; only letters Cicero wrote to Atticus. I guess that Atticus did not make copies of letters he sent, but did keep an archive of letters received. Atticus died in 32, long before the letters seem to have appeared. However, Agrippa married Atticus' daughter Caecilia Attica. Their daughter, Vipsania, was Tiberius' first wife. Agrippa's third wife, Julia, was also Tiberius' second wife. The letters, if they stayed in the family, might have passed from Atticus' archives through Attica and Vispania or Julia to Tiberius, and ultimately into the Imperial archive. I don't think we know about the precise circumstances of their publication.
Once in circulation they might enter private archives – again copied by specialized slaves – sold in bookstores, or even libraries. Augustus began a significant library at Rome, there were others. That any survived until today indicates either extreme luck or continued literary interest in them.
I refreshed my memory with help from Brill's New Pauly and The Oxford Classical Dictionary, 3rd ed. I've read a lot by and about Cicero over the years.
Also, if you're interested in the kind of day-to-day logistics of writing and sending letters, you may be interested in this old collection of Cicero's comments about the mail I wrote some time ago.