great question!

interestingly, we believe ancient Rome was primarily an oral culture. meaning that the transmission of knowledge was most usually done through public lectures, demonstrations, or semi-private/private readings. Galen, for example, goes on at length about the public dissections he performs for large audiences, comprising of Romans of all classes. one Roman nobleman was so impressed with his knowledge, he asked that a copy of his lecture notes be made for him at once.

so, poetry, history, letters, etc. would have been read aloud within public spaces like fora, or within private/semi-private homes for the elites. this performance within private homes would have been important, as writers in ancient Rome most usually had a patron. Horace, for example, was patronised by Maecenas, a close friend of Augustus. this patron-client relationship was reciprocal. Maecenas assisted Horace financially and socially by integrating him into the elite circles of Rome, where his poetry would be more widely consumed. in return, Maecenas increased his social prestige by being seen as an ally of the arts and as a civilised and cultured man (among other political benefits unique to the time period). the text would then have been copied out, if a listener desired a copy for himself/herself. this process of oral transmission and written copies was how libraries like that of Galen flourished in ancient Rome. such libraries would have been stored in temples, the texts within loaned out to those interested parties who wished to make copies of their own. the destruction of Galen's library in AD 192 was a terrible blow to the physician!

for more on this, I would highly recommend Education in Ancient Rome by Stanley Bonner. Empire of Honour by Lendon is also a great resource for the patron-client relationship in Rome.