Pretty good question, and I've wondered this one before myself.
Frankly I think that 'eating' the soul is an expression that's as close to what the demons actually do as possible. This is on a scale that isn't obviously seen by us, so we wouldn't understand... From what I gathered by what you said, I'd theorize that the souls could never truly vanish or be destroyed, which is why the demons hunger for it, it might be like a battery that keeps them going for eternity. If they 'hunger' for it, then they need it to survive, in my mind. The more souls they gather, the more powerful they get, which might imply why demons like Sebastian are so powerful, since I have a feeling that he's done contracts many times before.
So, maybe through the eyes of the soul, they sit in something like limbo, wherever the demons might place them? Or they sit inside the actual demons themselves, which would just to them look like a blank space of nothing..ness? I'd imagine the demons could do whatever they want to them, having all of their ravenous fun, and when they're busy the souls just sit there, that's not the worst hell could be but it is definitely not heaven...
Generally, all I know for sure is that a demon wins when they eat you.