Is Sauron Frodo's shadow?

[Pitchwife]

More nice comments by everybody. We certainly don't have to keep this focused on Sauron and Frodo exclusively, and I agree that Gollum makes a much better example for D's theory than Sauron - Frodo under the influence of the Ring is in serious danger of becoming Gollum, much more than he is of becoming Sauron; which is precisely what enables him to understand and pity Gollum.
About the character development thing - yes, I think D has a very strong point here. Generally, in the literature of adventure, characters are studied and developed through action rather than introspective analysis, but in fantasy, everything that happens, everything a character does or encounters is, or can become, characterization - even scenery: think of Aragorn under the Argonath, or Gimli and the Glittering Caves.
Boro does a nice job applying the theory to Boromir (as could be expected), and I'm sure this could be done for other characters as well.

Morth, while I agree that Frodo certainly didn't lust for power the way Sauron did, I'm not so sure he utterly lacked a will or need to dominate - nor the capacity, even though in him this was much more limited than in Sauron, or e.g. Gandalf or even Aragorn. There are a few interesting moments in Frodo's interaction with Gollum, starting in Book IV, The Taming of Sméagol:

For a moment it appeared to Sam that his master had grown and Gollum had shrunk: a tall stern shadow, a mighty lord who hid his brightness in grey cloud, and at his feet a little whining dog.

A little later (The Black Gate is Closed):

"In the last need, Sméagol, I should put on the Precious; and the Precious mastered you long ago. If I, wearing it, were to command you, you would obey, even if it were to leap from a precipice or to cast yourself into the fire. And such would be my command. So have a care, Sméagol!"

And finally (Book VI, Mount Doom):

Then suddenly, as before under the eaves of the Emyn Muil, Sam saw these two rivals with other vision. A crouching shape, scarcely more than the shadow of a living thing, a creature now wholly ruined and defeated, yet filled with a hideous lust and rage; and before it stood stern, untouchable now by pity, a figure robed in white, but at its breast it held a wheel of fire. Out of the fire, there spoke a commanding voice.
"Begone, and trouble me no more! If you touch me ever again, you shall be cast yourself into the Fire of Doom."

I don't doubt that Frodo, wearing the Ring, could have dominated Gollum in the crucial moment, if he hadn't been taken by surprise. By the way, whose voice is speaking here, out of the fire - Frodo's or the Rings? They have become indistinguishable.

Morsul lays his finger on what I feel to be the main flaw with applying D's theory to LotR (literally, at least):

That is to say LOTR only happened in Frodo's head

This is true for D's Covenant, who is unconscious or dying in the 'real' world every time he's summoned to the Land, so everything that happens to him there may well be 'only in his head' (not that this makes a difference, as he finds out). In LotR, however, the division between a primary and secondary world doesn't enter into the story, so both Frodo and Sauron, as well as all the other characters, are on the same ontological level, denizens of the same world. But I think the theory can be modified to fit.

(This is going on really nicely, thanks to everybody for contributing!)