This is a two in one question:
Should the asker of a duplicate question gain reputation for asking the duplicate question?
Should the answerers of duplicate questions gain reputation for answering them?
By asking a duplicate question you are diluting the value of the site. It takes lots of TLC to find the dupe and merge it in, yet you are still rewarded for producing that work.
The answerers of the duplicate question are also diluting the value of the site, because good and valuable information becomes splintered and harder to find.