Each verse is basically:
They say that ev'ryone
Needs someone
So how come no one
Needs me
with later iterations replacing "needs" with "wants" and then with "loves." (Compared to the Everlys' version, the Beatles flip the order of the first two verses. The Everlys sing "needs" and then "wants"; the Beatles sing "wants" and then "needs.")
There really isn't a rhyme scheme here; it's just "-one" rhymed with itself, which illustrates the speaker/singer's loneliness and solitude. It's just "one," "one," "one."