So I looked at the magic item description and the page for it on the Forgotten Realms wiki, and nothing in either location says that the magic in it is explicitly used up and can't be used again. So could you theoretically receive a message from a Paper Bird, read it, write something else on it, and then send it back?
No, it is not reusable.
The description of the Paper Bird, as published in Waterdeep: Dragon Heist (pp. 191-192), states:
It travels to within 5 feet of its intended recipient by the most direct route, whereupon it turns into a nonmagical and inanimate sheet of parchment...