Ecce Romani II, an intermediate latin textbook, describes the postal system in the time of the Roman Republic as almost entirely private, but with a communal twist. The two major methods for a private citizen to send a letter across the empire would be to send a slave or other courier or, more interestingly, to find a passing traveler heading to the same destination as your letter and ask him or her to forward it along for you.

Ecce Romani depicts this sort of reverse-hitchhiking as part of the social mores surrounding hospitality towards guests- just as a Roman was expected to offer some measure of welcome to passing strangers, travelers were expected to occasionally carry messages for their hosts.

edit: This doesn't have to do with the Romans, but one of the most famous descriptions of postal services in antiquity comes from the Histories of Herodotus, describing the imperial postal service of the Fifth-century BC Assyrians:

"While Xerxes was doing thus, he sent a messenger to the Persians, to announce the calamity which had come upon them. Now there is nothing mortal which accomplishes a journey with more speed than these messengers, so skilfully has this been invented by the Persians: for they say that according to the number of days of which the entire journey consists, so many horses and men are set at intervals, each man and horse appointed for a day's journey. These neither snow nor rain nor heat nor darkness of night prevents from accomplishing each one the task proposed to him, with the very utmost speed. The first then rides and delivers the message with which he is charged to the second, and the second to the third; and after that it goes through them handed from one to the other, as in the torch-race among the Hellenes, which they perform for Hephaistos. This kind of running of their horses the Persians call angareion." -Histories, Book 8 verse 98.