If enough people upvote the link to the originally posted question, or even if it gets selected as the answer, then SO can simply _not index_ this question. But the poster doesn't feel rejected, and can in fact choose to continue engaging in his question in cases where it's subtly different.
Sometimes the duplicate is blatant, but in many case it is really a judgment call. In my experience I more often vote for "reopen" than for "close", so there are definitely different attitudes to what is considered a duplicate.
Alternatively, it could be the decision of the one asking the question. Just as they select the "accepted" answer, they could choose to accept or reject the duplicate as appropriate.
Here's something I've never really understood: What is the downside to leaving the question up, even if it is an actual duplicate? Instead of "closing as duplicate", just answer it with a link to what you feel is the question it duplicates. If that works, then the original will score higher on Google, and if it turns out it's not a duplicate, the asker is able to provide more information as to why that answer doesn't work for them.
The asker can edit the question to provide more information and the question can be nominated for reopening. It only requires 5 votes to reopen, IIRC.