Obviously the Athosians find out pretty quick due to their situation of living with the Expedition at Atlantis. I can't really remember if they simply tell the Asurans who they are or if they find out from the hand-in-head thing. The Wraith... I'm not sure if they know straight from the beginning that Earth is in another galaxy or if they just know that it's somehow unknown to them and the rest of Pegasus. They certainly know at some point that it's not in Pegasus, even if they don't know it's in the Milky Way (otherwise the Superhive in the finale wouldn't have needed the signal from an alternate reality before heading there).
But what about everyone else in the galaxy? In 'Inquisition' - episode 5.13 - the Coalition folks seem to know. They say things like "since your arrival in this galaxy." I don't remember the Expedition exactly advertising that though. I think I can remember them being rather vague about stuff, saying that they "weren't from around here" or whatever. Now, I can certainly see how that information might leak out over time. An Athosian says something innocuous to a trader, Michael says stuff about it to his Hybrids, the Genii got some intel from when they took over Atlantis, things like that. But I certainly can't remember them telling someone outright and the other party acting suitably impressed; it is one thing to walk to different planets through a magic ring left by the Ancestors but it's quite another to be able to fly in spaceships from another galaxy.
I don't think it ever was made explicit, I just wondered if I missed something.