In the old days, players had to consciously opt-in to be eligible for the draft, so I'm sure they took the initiative to follow the draft results. Now that it's an opt-out system, I suppose it's possible for someone to remain ignorant of their new draftee status.
But virtually every team contacts its draftees to invite them to some kind of prospect camp. So unless the player lives in the middle of Antartica, he won't stay ignorant for long.
I can't remember exactly who it was, but I remember someone was working as a busboy at a hotel and were notified that they had been drafted several months afterwards so he just left the job right then and there.
Dominik Hasek said he didn't care about the NHL draft at all, he found out just because his teammates were searching the papers for their names and told him that he was drafted too. (I hope it's correct, it's a few years since I read his autobiography).