There was no Roman mail system. During the Principate a system of official couriers was set up over time, but they delivered state and military correspondence. Private individuals, in the Republic as in the Principate, delivered mail by private means. Individuals had to make their own arrangements for the delivery of mail, which might mean sending out a slave from the household or (if one was a sitting magistrate) possibly even sending magisterial staff out. But usually letters were delivered by travelers going in the right direction that the sender contracted to deliver their letters or tablets for them. In some cases this was fairly well-organized (publicans often carried letters on their way to the provinces, or employed staffs of tabellarii, private letter-bearers, usually slaves, to do it for them) but even for the wealthiest individuals quite frequently letters had to be entrusted to whatever traveler happened to be passing by, heading to the right place.

Because there wasn't really a postal service at all, naturally letters might vary in the time of their delivery. State and military correspondence, which could rely on established, maintained courier routes, could arrive very fast indeed, and professional couriers like tabellarii could make pretty good time, especially over land. Relying on travelers could take some time, though, especially if the traveler was not using the fastest route. Also to be considered is that the fastest means of travel over long distances in the Roman world was almost always by sea. However, sea travel was pretty irregular--given good winds even the longest distances could be traversed shockingly quickly, but in unfavorable conditions sea travel might only be marginally faster (though usually much more direct) than travelling over land.