Originally Posted by
KOUNTERPARTS
I would assume the general populace of Azeroth -- not the heroes (us) or important lore characters -- are aware they are dead. At least... dead in the common sense.
Personally don't think they are aware of the aspect of Old Gods either A.) not truly dying, B.) Old Gods being so rooted into Azeroth that killing them is a bad idea, or C.) some of them (Yogg-Saron) seemingly returning.
The way I see it, they were aware of the first initial menace the Old Gods caused at their time of being a menace. C'Thun back during vanilla timeline and the war effort in Silithus, Brann and Rhonin declaring the problem at Ulduar with Yogg-Saron, and then N'Zoth's assault on Azeroth which is known to the entire planet. All of those respective times we have come out the victors so I would think the general population just thinks that: the Old Gods are defeated. Whether that means dead, dormant, whatever.
TL;DR: I don't think the common person of Azeroth is aware of the specifics of Old God lore themselves. Ignorance is bliss.