Although the fates of Saruman and Sauron are not revealed, I believe a central assumption can be divined for both of them. Tolkien may have eschewed allegory, but his symbolism is quite apparent.
First, Saruman's demise:
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…about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a cold sigh dissolved into nothing.
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Now, Sauron's exit:
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...there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was blown away...
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The emphasized phrasing in both quotes are mine.
Although it is clearer in Saruman's case (particularly when Tolkien refers metaphorically to the grey mist appearing like a pale shrouded figure gazing almost imploringly to the West), it is plain that a great wind, like the pounding of a judge's gavel, passed final judgement on both of them and dispersed their spirits forever. Unlike Morgoth, whose spiritual and/or physical manifestation was imprisoned until the final battle at the end of all things, Sauron and Saruman would remain incorporeal and impotent, and would no longer plague Middle-earth.
That a "great wind" or "cold wind" should propitiously be summoned at such precise junctures (a 'cold' wind in Saruman's case, because he had betrayed his sacred trust) indicates, in my mind at least, the intervention of Eru, or perhaps Manwe, because "the winds and airs were his servants, and he was lord of air, wind, and clouds in Arda."