Does "Either must die at the hand of the other" imply both that A can only be killed by B and that B can only be killed by A?

As a pronoun, either has a meaning of "one or the other of two people or things" (Lexico). In the sentence, the contrast is between one of them and the other one: one must die at the hand of the other. Logically, this gives two mutually exclusive possibilities:

• Voldemort must die at the hand of Harry
• Harry must die at the hand of Voldemort

Once one of these possibilities happens, the prophecy is fulfilled: either has died at the hand of the other.

The alternative interpretation you give that requires both possibilities to be true would require a different initial pronoun, e.g. both:

• Both must die at the hand of the other.

Or better yet:

• Both must die at the hand of each other.

In math logic, A OR B statement can True in three cases:
(1) A is True and B is False
(2) B is True and A is False
(3) A and B is True

If “Either” means: one OR the other of two people or things, then "Either must die at the hand of the other" could be true in the case that both will die at the hand of each other. This is similar to the case (3) above. This prophecy admits a few ways of being fulfilled but in the HP series the third option is chosen.