We had a hint that Quirrell is in some way Voldemort, but how exactly? We haven't been given information from anyone but Quirrell on that. They went to the same dojo. Physically, that's nearly all we've been told, by an unreliable source. The only other physical hint is Quirrell's zombie/alertness transition.

The questions are these: Are they two distinct individuals? How distinct and why? Have they always been so or not so? And is each aware of the other?

Here's another idea, no more motivated than the idea that Q=V. Perhaps when "tuning in", Quirrell is not channeling Voldemort. Rather, Quirrell has a "mysterious dark side" which feeds him ideas when he is in zombie-mode. Neither the former Quirrell, or the remains of Voldemort were powerful enough to impact magical Britain sufficiently to satisfy their goals. Additionally, their goals did not match perfectly. A trade was made; a partial exchange of utility functions, and perhaps something more.

Voldemort's magical knowledge was vast and useful; if someone thought himself to be terribly clever, and thought that he was less of a fool than the now-failed Dark Lord, and sought power, he might make a deal with the devil; confine the mind to a tightly sealed box instead of destroying it. In exchange for this life-saving act, knowledge. And with every exchange, there could be some influence on each other's values. Maybe he's already admitted that he's going to lose - the mind will get out of the box by the end, and he will cease to be distinguishable. But as a new challenge, he bet he could transfer the best of his knowledge and goals to Harry Potter before then, enough to defeat whatever would grow from his mental corpse.

Or not. All sorts of things are possible with the little information we have. The most mysterious repeating phenomenon that's notable right now is Quirrell's zombie-mode. For just his political views and most of his behavior, being Voldemort is not the simplest explanation.