Okay, ready to try this out...
First and foremost, I feel that a story's meaning comes from the author's intent. As I'm pretty sure I've seen stated in other threads, the author ultimately has ownership of any story as its creator. Expanding on
Wilwarin's example, with an author kind enough to provide commentary:
Quote:
"Johnny killed Bob and then went to jail" is a pivotal statement in my novel Kiss the Moonbeams Off My Elbow. In the context of Johnny's struggle against society's injustices as represented by Bob's character, Johnny's sentence is unfair and perpetuates the endless cycle of cruelty Bob embodied.
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Once this statement has been made, it's pretty hard for a reader to argue that the real meaning of Bob's death was to show that Johnny was a ruthless criminal who deserved to go to jail for Bob's murder. The author told us exactly what was meant in the situation, even if the reader disagrees. But what if the statement had been something a little different?
Quote:
All novels must have enigmas, even crime thrillers, and Johnny is one of them. His state of mind when he killed Bob is certainly a mystery, and it was a mystery to me when I wrote it. I suppose he was meant to behave as he did; yet I cannot find a good explanation for my choice to write that passage as I did.
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Or maybe there are other books in the series that contradict each other as to Johnny's motivation.

In any of those cases, the reader's collaboration with the author fills in the blanks and becomes part of the meaning of the story. When there is ambiguity, I think there is room for extra interpretation.
Furthermore, I think that any story derives extra personal meaning from how we react to it. It may not be the real meaning if there are statements from the author that contradict it, but it is a meaning, nevertheless that has value to the individual reader. In the case of the LOTR, I think that we have additional permission to look for our own meanings in the story due to the author's dislike of allegory and preference for applicability. By bringing the word applicability into play, I think we're invited to look for how the story relates to our experiences and into collaboration with the author. But there are limits within the rules of Middle Earth. A theoretical reader can't choose to interpret Sauron as Tolkien's great hero because of sympathizing with his lust for power and plans for complete world domination. That flies in the face of the author's intentions. But it is entirely legitimate to debate whether or not Sauron had a chance at redemption.
So I suppose my position boils down to giving the author the last word, especially when there's a clearly stated intention, but also giving the reader space to interpret and react to the story. I hope it's not too contradictory.
EDIT: I almost voted for "all of the above" but I don't think I can give the reader the last word for the real meaning of a story. If the question were simply meaning, then I would have chosen that option. Any reaction can be meaning if significant to the reader, but I'm not sure that I can give all of them equal weight (as in the bizarre examples above).