There was a Roman "office" or occupation devoted to scribing, appropriately called a scriba (plural scribae). Our sources do not indicate any sort of professional requirements at all, and it seems that none existed formally. One could be nominated to the position by a higher official, but a scribe post could also be purchased like a modern-day taxi medallion. So the poet Horace, when a young man down on his luck, had purchased a spot among the scribae before his poetry catapulted him to fame and riches. The scribe's post was vaguely working-class, and our sources often refer to it with a bit of a sneer. There were basic requirements of legal status and moral suitability (no liars or rogues allowed!), but there is no evidence whatsoever of any professional requirements, and certainly no indication of any training. The basic requirements for the job must have been obvious: writing proficiency in a variety of forms, elevated reading proficiency, and basic arithmetic, all of which would be required for the scribe to do his job: qui...in rationibus monumentisque publicis versantur ("those who are involved in the public accounts and records")(Cic. dom. 74). Studies of Roman accounting and record-keeping have suggested that it was astonishingly rudimentary, considering the scope of of the Roman state by the Late Republic. By contrast, accounting and records in the private banking sector were a whole other universe of complexity. This is in keeping with the general narrowness and simplicity of the Roman bureaucratic apparatus in general, especially in the Late Republic and Early Empire, when a ludicrously small body of officials administered an entire empire. Bureaucratic complexity increased in the later Imperial period.

That there was no obvious training regime for scribae is not at all surprising. There were no standardized educational protocols for any occupation, from sandal-makers on up to legionary commanders. The standard was ad hoc apprenticeship, and it was a similar path for the lowliest laborers all the way up to the loftiest senators-to-be. Elite Romans might go abroad to study philosophy and rhetoric (like Cicero) with individual celebrity scholars or at "schools," but their real training was with an older mentor, either in the courtroom, the senate-house, or the battlefield. Even for highly technical fields, like the enigmatic military engineers, we have essentially zero evidence of formalized training. I talked a little bit about the lack of obvious training apparatus for engineers over here.

Numerous inscriptions record the epitaphs of those who called themselves scribae, and from them we get a sense of divisions within the profession. Hartmann (The Scribes of Rome Cambridge 2020) has a big appendix which gathers them. Some were attached to specific offices, like the quaestors or aediles. So a L. Aelius was scriba questorius sex primus curatorum (Hartmann #1). P. Aelius was a scriba tribunicius maior (#2, attached to a tribune, perhaps a military tribune?), and also a scriba aedilium / quaestorius. Most seem to have belonged to groupings termed decuriones, the same word used for cavalry squadrons. So C. Aelius was scriba librarius quaestorius trium decuriarum (#4). A guy named Faberius calls himself scriba but also γραμματεύς, presumably the Greek equivalent or an adjacent role (#68). Nampius was a scriba officii viri clarissimi legati almae Karthaginis, clearly a scribe attached to the governance of Carthage after that city's refounding as a Roman colony.

All of that pertains to public scribes. There were also private scribes, like Cicero's secretary Tiro. They are often termed librarius, but the monikers are loose. They also, of course, had no formal training that we can tell, though often they were Greeks (like Tiro) and presumably had received an excellent formal education in their former lives (like Tiro). Many were also slaves (like Tiro). Public scribes did not busy themselves with the copying and disseminating of books—this was all private enterprise. Elite households would copy books, and there is some very thin evidence of "copy-houses" for the purpose, probably owned by individual Roman aristocrats. There is also some evidence for larger libraries (like the libraries of the Imperial fora) doing this. I'm not so sure about that aspect, however, and hopefully someone else will have more insight about the private sphere.