Well, if you're living roughly 80 miles north of Hadrian's Wall, you might in fact be a Roman citizen. Roman control over the region north of Hadrian's Wall was frequent for most of the Roman era in Britain; Agricola and others expanded the frontier to the Scottish lowlands in the late 1st Century AD, and when Hadrian's Wall was built, it was well within what you might consider the "Roman border". The later emperor Antoninus Pius built an additional wall further up the peninsula in the 160s/170s, of which only the foundation survives, and Septimius Severus (d. 211) was in Britain at the end of his reign, planning to push even further north.

This is, of course, complicated by exactly what a "border" meant in the premodern world; you might imagine it as a hazy "zone of control" where individual communities knew which tribe they belonged to and whether it was ultimately under Roman control. The geographical limits of that control might not be clear, and might have been more closely linked to tribal ranges or roaming patterns rather than fixed lines on a map. It's certainly possible to imagine people in rural regions or at the fringes of the Roman world not "realizing" that they were Roman or not. This is because the interaction of the average citizen with Roman cultural elements or social control were heavily interlaced with urbanism and social status: if you lived in a city, or were wealthy, you had access to more "Roman" cultural matter, such as social institutions (forums, public baths, magistrates, public entertainment, formal law). Peasants may have simply paid their taxes to their local chief or magistrate, worked their farms, and never seen anyone in Roman dress or heard the Latin tongue.

Now I know your initial question was probably asked with the perspective of the local peoples, the Picts or Scots, in mind. Their history is much more difficult to reconstruct because they were both preliterate and only infrequently mentioned by Roman written sources of the period (our best source for Britain at all is Tacitus' Agricola, which is from around 100 AD and only somewhat reliable for ethnographic information). Once again, a defining factor would be social position. A chieftain of the Picti would definitely know who the Romans were and interact with their local officials on the subject of trade relations, border crimes, or mercenary hires. Many of them were at least moderately peaceful client states (de iure or de facto) of the Romans. An average person among the Picti who didn't travel much might only know the Romans as the people who live "over there" in a vague sense - much as, as a Californian, I have a vague sense of the geography and people of Arizona. But, unless there was a war on, he or she would be unlikely to have ever met one (except perhaps a trader) in the flesh.