So there's a few different tasks that could be thought of as librarianship. At the lowest level were the official scribes, responsible for copying each work for dissemination. Supposedly, the authorities confiscated all original texts that entered the city and then returned copies to their owners.

Then you've got the scholars, people came from all over to learn from the texts stored there. Euclid is probably the most famous, but there were quite a few other significant scholars who spent time there.

Then you've got those who are more traditionally considered librarians. They mostly doubled as scholars, like Eratosthenes, who found the circumference of the earth while head librarian. Due the influx of so many books, organization was unsurprisingly a large problem. Zenodotus, regarded as the first chief librarian, arranged all the books in alphabetical order by author. He was also a noted scholar of Homer and distilled the various versions of the poems into a single edition. One major decision a lot of these librarians were responsible was determining the accuracy of various works and removing lines that they felt weren't in line with the original text.

Callimachus, one of the later librarians, produced the Pinakes, a 120 book compilation of all the books in the library, sorted by author/field of scholarship.

The librarians were also responsible for the development of early punctuation. Aristophanes developed various dots that became the comma and the period. Aristarchus developed the asterisk. Both were also renowned Homer scholars.