I just completed a master's thesis (literally yesterday) in which I argued that Rome constituted a nation by the third century BCE, and it continued down a path of ever growing nationalism through the Social War of 91-89 BCE which resulted in the enfranchisement all Italy into the Roman legal system. It ends with Augustus formally unifying Italy upon his ascension as first emperor after 27 BCE.

I state all this because the central focus of my argument was that your average freemen, or even slave in Italy had to be aware of what other Romans and Italians were doing at any given time in order to create a sense of community. Benedict Anderson refers to this as an "imagined community," which in short means that you, as a citizen of whatever country you are from, holds a sense of simultaneity and calendric progression with your fellow citizens, even though you will never meet them all. So an American might feel full confidence in stating "Americans are all celebrating the fourth of July right now, eating hot dogs and watching fireworks." This is the essence of simultaneity and imagined communities, that a guy in California knows a girl in Rhode Island is participating in the same event.

How does this fit into Rome? Well, you have focused solely on the imperial period, but as noted, I began looking at Roman behaviors from the fourth century BCE down through the first century BCE, ending my thesis with the first Roman emperor. Rome in this era did not encompass all of Italy, instead if had forged a very small nation comprised mostly of Latium (surrounding area) and parts of central Italy that included the Etruscans, this in addition to their colonies of citizens.

Anyway, we have these religious and political calendars chiseled in stone or bronze, a few painted, called the Fasti. The Fasti tracked religious holidays and followed the election of various magistrates. There have been 47 (I think) total Fasti recovered in Italy and over 25 are from Italy but outside Rome. The fascinating thing we have found is that your average freemen, and even slaves in some cases, had formed groups to create their own Fasti. These calendars sometimes follow the religious festivals in Rome, and the magistrates elected, even in cities miles away from Rome! Thus an inhabitant of say Tusculum, could know precisely what the people in Rome were up to on any given day. The Fasti were also sometimes hung publicly, meaning the entire village could know what Rome and any other city under the Roman umbrella was up to.

To this we can add coinage as a means of spreading news, of which Carlos Noreña has done some outstanding research. He found that coins were one of the best forms of mass media of the ancient world. Rome was so dominant, that their centrally produced coinage touched the entire Mediterranean world, and when issued new to the military (where most of it went) it featured the latest imagery and propaganda of Rome. Who the important leaders were (or emperors in later periods), what temples and religious festivals were important, who won a war, what people were conquered etc. The problem with coins is they can stay in circulation for a while, so messages get old, but high circulation of new coinage to the military, especially in the later empire, got messages to frontiers in mass way.

We must also consider the role of public architecture and inscriptions. Upon Augustus' death, these massive inscriptions known as the Res Gestae went up all around Italy and the empire, the self written deeds of Augustus' life. Some of the best preserved ones we have come from Asia Minor (modern Turkey). They speak of how he defeated various enemies, rebuilt 80 temples, expanded Rome's dominion, he also mentions many specific state leaders, Roman gods and on and on - read it all here.

The Res Gestae was installed everywhere, in public centers, near religious temples, across the entire Roman world. Granted literacy rates of the era have been estimated at 15-20%, but both Suna Guven and Azar Gat argue that basic literacy was more widespread (people who could read a little) and there were public readers who recounted the information on these inscriptions. So some city dwelling peasant, walking around Asia Minor one day, might hear the marvellous tales of Augustus and gain a perception of being part of a much larger world, one that he could also see on the currency he used.

There is much, much more to this, but in short, yes, many people (can't claim all) living in both Italy outside of Rome and in the empire had some level of knowledge about the wider world. As the empire expanded, this becomes more difficult to assess, but I will perhaps address it in a later work.