I suspect that Paul wrote letters because he was responding to the same social and political pressures that the Epicureans were responding to, whether or not there was any connection between the two traditions--both groups were barred from teaching in the Gymnasia, the Agora, and the Roman Forum. So they found recourse in one of the only options left to them. This was a stroke of great luck for us, as Zeno's whole output from the Stoa is lost except in fragments while Epicurus' letters survive.
In the Renaissance and the Enlightenment, a similar phenomenon developed called the Republic of Letters, a complex web of private correspondence for the transmission of ideas. Ironically, the term was first used by Francesco Barbaro in a letter to Poggio Bracciolini, and in the very same year--1417--that saw the rediscovery of Lucretius by the latter. In one surviving letter, a friend of Poggio pleads with him to be more guarded in his writing--a letter stops being private if it falls into the wrong hands, after all.
Poggio's letter written in admiration of Jerome of Prague, a heretic murdered by the Church at the Council of Constance, is here.