Burney describes Neapolitan conservatory students "pounding out" accompaniment exercises that were full of "jargon and dissonance" in his history of music and Robert Gjerdingen suggests that these exercises were partimenti. Assuming that Gjerdingen is correct, does Burney's statement imply that students did not get the partimento down perfectly at the first go, but rather would have to work at it a bit? If this is the case, would teachers have occasionally had students write out realizations to better understand what they were playing?
The book "The Art of Partimento" by Sanguinetti contains a history of partimento, and an exposition of the practice, by way of the author's study of the subject, and through direct translations of the original Italian (Venetian) manuscripts on the teaching of partimento. It may have the answers you seek. It is, in my opinion, the best musical text one can find.