"Literacy was not widespread in Mesopotamia. Scribes, nearly always men, had to undergo training, and having successfully completed a curriculum became entitled to call themselves dubsar, which means 'scribe'. They became members of a privileged élite who, like scribes in ancient Egypt, might look with contempt upon their fellow citizens." Source: Ancient History Encyclopedia.
Note: In the beginning, Mesopotamian scribes were primarily involved in the recording of financial and government transactions and they were usually anonymous, but this changed as time went by and and the first literature began to appear.
"Some scribes signed their names to tablets as early as 2600 BCE. Generally, the authors of literary texts remained anonymous; however a catalogue from Nineveh listed authors and editors of some well-known compositions such as the Epic of Gilgamesh. Later scribes listed their lineage, giving the names of their fathers and even earlier ancestors." Source: Daily Life in Ancient Mesopotamia, by Karen Rhea Nemet-Nejat.