This question seems to have been asked quite a lot, but I couldn't find much in terms of answers.
This post by u/RotNS suggests that the question is difficult to answer due to the availability of "average person" sources. I'm sure there's a lot more to be said on the topic, but I wasn't able to find any other previous responses that really answered your question.
In addition to the link by u/Supertigy, I've previously compiled a list of some other answers to similar questions:
How confident are we that the year is actually and exactly 2016? Is it possible that at some point in the last 2000 years there were any significant timekeeping mistakes? and Did people in the Middle Ages call the year "one thousand and one" or "ten oh one"? by /u/sunagainstgold
At what point did society begin referring to the year in modern terms (2012 etc)? by /u/Algernon_Asimov
Was the millennium from 999-1000 ad marked in any special way? by /u/savvysioux
Did people in 999 AD celebrate the new millennium? Were there any doomsday predictions similar to Y2K? by /u/haimoofauxerre
And a couple by me:
Did the Eastern Roman Empire date by Anno Domini or by Ab Urbe Condita?
In brief, the answer is, as Supertigy mentioned, we really have no idea what the "average" medieval person thought about anything, including what year they thought it was. But among educated people, there were numerous different kinds of calendars, depending on what they were writing about. Latin Europeans knew about the BC/AD dating system devised by Dionysius Exiguus, but for official documents, they tended to use "regnal dates" instead - for example, if it was the year that we now consider to be "1200", they might instead write "in the second year of King John" (in England) or "in the third year of Pope Innocent III" (in papal documents).
Meanwhile in the Greek-speaking world they preferred the "anno mundi" calendar, dated from the creation of the universe, which they established as 5509 BC. So the year 1200 on the anno domini calendar was the year 6708.
Likewise, Muslims used a calendar based on Muhammad's migration to Medina, which was 610 AD, so the year 1200 AD was the year 590. Medieval Jews also used an anno mundi calendar but with a different calculation than the ones the Greeks used, so the year 1200 AD was the year 4960.
But again, the "average" person may or may not have conceived of the calendar this way, we really just don't know.
As others have said, the paucity of sources of “average” voices from history make this an almost impossible question to answer. And though it may not be applicable across the board, I do think Richard Landes’ article about millennialism might be instructive here. Against contemporary scholarly consensus, his article The Fear of an Apocalyptic Year 1000: Augustinian Historiography, Medieval and Modern serves as a return to the idea that the years leading up to the end of the first millennium were marked with fear and unrest.
What is of interest to us here is his discussion about the sabbatical millennium, or the 1000 year kingdom of heaven. Pulling from scripture early clerical leaders combined the 6 days it took God to create the Earth with the passage “a thousand years is as a day in the sight of the Lord” (p. 110) to create the expectation that the second coming of Christ would occur 6000 years after creation, tracked as annus mundi (A.M.). This was a useful tool for clergy to combat those who preached an impending apocalypse, since they could point to this calculation as proof that Judgement Day lay centuries away, claims that were confirmed as each passing prophecy did not come to pass. Importantly, this was not merely a scholarly exercise but served the purpose of pacifying the masses; Landes says “it was, above all, a teaching intended for those unsophisticated believers who were so easily swayed by apocalyptic rhetoric that they followed the delirious ravings of false prophets” (p. 111). That is, this measure of time was entirely directed to tamp down panic among the laity.
But once the highly anticipated date approached, instead of providing relief to those who might be swayed it threatened to upend social order altogether. And so we see recalculations in the fourth century that push the birth of Christ back from 5500 to 5200, thus pushing off the apocalypse for another 300 years. Landes then points to a series of similar behaviors in the ramp up to the new chronological 6000 AM (annus mundi ii) to lend credence to his belief in a popular apocalyptic worldview around these dates, including increased clerical denunciation of dangerous millennialism. As evidence of the widespread nature of the issue he, says “Bede even complained about rustics who pestered him with questions about the number of years remaining in the millennium. Such a rare and revealing remark indicates that commoners were attuned to the millennial countdown and that with the approach of 6000, such concerns could only get worse” (p. 113). The final chronological transformation, from annus mundi ii to annus domini, pushed the millennium even further out, and was notably adopted several hundred years after its creation when worries about an apocalyptic future reared up again.
Ultimately Landes says much the same as has been repeated here: we don’t know. There is not enough proof to give an indication of public conception of time. But reading between the lines it becomes clear that common people were aware of the greater thrust of time, that we have “evidence that there were ’pseudo-doctors who rose up,’ ‘renegade clerics who mislead the people,’ and wandering preachers and penitents ‘without any law,’ some of whom circulated dangerous ‘letters from heaven’ that ‘should be neither read nor taken up, but burned’” (p. 115).
https://www.bu.edu/history/files/2011/10/11.Fear-of-an-Apocalyptic-Year-1000-Speculum.pdf