Other than dwindling question quality, I'm not sure this is a problem.
There are countless Stack Overflow questions where someone asks an easy to answer question, and people respond with "Do the research yourself" or "the first Google result tells you". One way or another, the question and its answer become very popular and the question becomes the first Google search result. Thus, a question that is "low quality" by SE standards becomes the place to get the answer to that question. This question is one such example; it's really easy to find in many different places, but now the SO answer is easily the best one available.
Now, maybe programming issues lend themselves well to the question/answer format SE is going for, but maybe we can see the same phenomenon here. If we post "the same old puzzle" here, it might get oddly insightful answers from odd and insightful people. We might get people who link to similar questions and answers, or three answers that tackle the same puzzle three different ways, or to three levels of rigor and detail. All of that goes beyond "here's a riddle and there's the answer". Daydreaming aside, I think there's value in having it all in one place; it would be nice to think of a puzzle and think, "hey, I'll see if there's anything on Puzzling like this".
So if you see a repeat of a question on Puzzling, absolutely flag it as a duplicate. Draw the asker's attention to the original, so it can get more upvotes and become the place to look for such a puzzle. If you see a question that's just too simple or otherwise not high quality, "downvote it into oblivion" or flag it appropriately. But as far as I'm concerned, asking a question another site has asked (answered) before is a chance to ask (answer) it better.