When I asked this question, one of the answers, now deleted, was suggesting that the type Either corresponds to XOR, rather than OR, in the Curry-Howard correspondence, because it cannot be Left and Right at the same time.
Where is the truth?
The confusion stems from the Boolean truth-table exposition of logic. In particular, when both arguments are True, OR is True, whereas XOR is False. Logically it means that to prove OR it's enough to provide the proof of one of the arguments; but it's okay if the other one is True as well--we just don't care.
In Curry-Howard interpretation, if somebody gives you an element of Either a b, and you were able to extract the value of a from it, you still know nothing about b. It could be inhabited or not.
On the other hand, to prove XOR, you not only need the proof of one argument, you must also provide the proof of the falsehood of the other argument.
So, with Curry-Howard interpretation, if somebody gives you an element of Xor a b and you were able to extract the value of a from it, you would conclude that b is uninhabited (that is, isomorphic to Void). Conversely, if you were able to extract the value of b, then you'd know that a was uninhabited.
The proof of falsehood of a is a function a->Void. Such a function would be able to produce a value of Void, given a value of a, which is clearly impossible. So there can be no values of a. (There is only one function that returns Void, and that's the identity on Void.)