Tom's ability to see Frodo even when he's on 'the other side' only makes sense in light of what we've been saying about Tom here -- he is either a manifestation of (in a literary sense) or spirit from the natural world, so of course he would be able to see everything that exists in nature. If putting on the Ring placed one in a different natural order, then this would mean that Sauron had succeeded in an act of creation that was on par with Eru -- that is, in creating a 'world' of his own.
The fact that Tom can still Frodo when he is invisible to others is proof positive that the Ring is part of a perverted or subverted nature that Sauron has managed to twist to his own purposes, not the gateway to a whole other realm.
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Scribbling scrabbling.
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