In any case, while I think this is a good idea, I also think that if a person asks many questions and doesn't stay on top of them, it's sort of too bad if their question gets closed. Until one gets an answer they accept, they should curate their question and engage with those attempting to offer an answer. That's my view in general, but I am rethinking this for the duplicate closure case.
This special case of being closed as a duplicate is more persuasive: it functionally asserts that the question has been asked before and increases the probability that an answer will be found by the OP. It may or may not be true that an answer actually exists for the other question (and it may not be true that the question is a duplicate), but, if so, an answer to the prior question should be quite useful to the asker. This helps SO because others may see the duplicate and the asker may refine or comment on their question if they have been notified. The newer duplicate is what shows up higher in recent activity lists and may be followed by some, so it is wise to temporarily support engagement around it for a little longer in order to migrate people to the prior question.
I'll try to put it a different way: suppose that a person asks a question and it is closed for non-dupe reasons (e.g. non-constructive), their TTA (time to answer) is not affected - an answer might never come or might be useless (either to them or the community). In the case of a duplicate, there may be a question that's already answered. In that case, getting a notification is like getting a link to a successful query of SO or Google, minus the latency of posting a question. This reduces the OP's TTA. When a question is closed as a duplicate, it is useful if the OP and anyone following the question are notified - it will reduce TTA for all of them.
So, please send an email to the OP if the question is closed, though askers should remain vigilant on their questions anyway.
Update 1: Thinking about question curation and TTA a bit more, I realized that new users may not be very good curators. The number of people who visit SO and post 1 question they "really need to answer now" may be pretty high. They may be inexperienced searchers, too. Getting a message about a duplicate could decrease their TTA and improve their time until return (i.e. reduce it). It's not so much that they won't return, as they may be unaware that they could get notifications, but that a prompted return (i.e. based on a notification) is likely to occur sooner than an unprompted return. That could also improve their subsequent engagement with SO in any number of ways - finding a better question, finding possible answers, and much more. Although I don't think I've had a question closed as a duplicate, I've certainly been motivated to learn more about how to use the search tools when the "Similar questions" tab on the right started populating with related questions that I didn't find when searching SO. If I hadn't noticed the tab, my question could've been closed as a dupe.
