Hmmm, it probably depends on where the power source is. for the gods, I personally assumed the faith is what gave them their ability to do things, so when Eros went to different worlds, if he was there for a day or so, he'd be fine, but after that his powers would start to wane and possibly disappear.

Now, if someone has an ITEM that gives them their power, and they bring that, they can make it their choice of if that item still works or if they feel it would start to wane as well (I believe Jack Frost gave someone a magical gift that lasted only temporarily in another world)

All in all, I'd say it'd be fine for short periods of visits, but should they stay there, say a month or so or long visitation periods? They might feel their powers start to lessen. :>