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.