So first of all, there were probably not a lot of medieval Latin Europeans talking about the year as 1001. Although the Anno Domini calendar was formally long established by that point, and the occasional monastic or royal chronicler refers to it, well into the later Middle Ages people tended to speak of regnal or papal or even local years (in the fourth year of the current king's reign; the year the duke got married, and so forth).

That said, remember that medieval Latin numbers until the late Middle Ages were Roman numerals (and the use of "n" for zero was extremely rare). 1001 would have been written MI. Does that reflect how people thought it? That's impossible to say, but it seems likely.

The only other parallel I can think of actually comes from the medieval Arabic world. The text we commonly call Arabian Nights becomes 1001 Nights sometime around the year 1000 (its textual history is enormously complex, but earliest versions promise a plan of 1000 stories for 1000 nights). In Arabic, it's Alf layla wa layla, thousand nights and [one] night. Of course, Muslims were obviously not using the Anno Domini calendar in the year A.D. 1001, but it's at least a data point.