"In every situation, I think authorship has to be acknowledged -- i.e., finishers shouldn't pass themselves off as the original, but declare that they're trying to continue a tradition, help complete a work of art, etc."
I completely agree with Corvus as stated above, and I think we should also ask ourselves:
1/ Who benefits by leaving the story unfinished?
2/ Who benefits by finishing the story?
3/ Who is hurt by leaving the story unfinished?
4/ Who is hurt by finishing the story?
To me it seems obvious that the reader benefits with the finishing of the story, and that is probably who the writer was trying to benefit in the first place, and that nobody is hurt by finishing the story, not even the author, as long as the above recognition is given. The whole thing shifts perspective if we are talking about a commercially published work, but those questions, honestly answered, should be a good guide.