In the early Roman Empire, Emperors would attempt to maintain the illusion of living in a Republic although they held all the real power and the position of Senator was mostly for social prestige, not for influencing the Empire politically. This was referred to as the Principate, and lasted until roughly 280 AD.

Later, that began to change under Diocletian, beginning the Dominate, in which Emperor's began to exercise their political power fully without really attempting to preserve the illusion of a Republic. This lasted until the fall of the Western Roman Empire in 476 AD.

Well, for one thing we should recognize that there isn't a dichotomy between republic/empire, rather between republic/monarchy. The imperium Romanum existed well before Augustus.

That aside, yes, ordinary people would have been well aware that there had been a drastic change in regime. The most obvious form of this would have been coinage, which is the most prolific form of political communication in both the ancient and modern world. Republican coinage, by and large, did not have the image of living people or current leaders. Augustus struck coins with himself in portrait--it is worth noting that Caesar and Mark Antony did as well, so this wasn't exactly unprecedented, which is also sort of the point. Other things Augustus would be doing would be making massive donations to the city, holding offices far more than he should and being followed around by the Praetorian Guard. Augustus gets a lot of credit for his constitutional settlement, and he deserves it, but he wasn't being subtle and he wasn't fooling anyone. Rather, it was a compromise of power and honor sharing.

so one of the key ways the "average joe" in the Roman empire would recognize the change would be in imperial iconography/propaganda. The deification of emperors, printing of coins with their names and mass building projects would give provincials a sense of who was in charge even if their personal lives didn't chanage very much. I would argue /u/aeronautically is probably overselling a pricipate/dominate distinction. Those two things are real and significant but your question of Empire versus Republic is too basic for this distinction to matter (especially since the imperial propaganda was the most important thing and that hardly hides the fact one man is ruling the empire). The illusion of the republic was long gone by say even the year of the five emperors