A moderator brought this question up a couple of weeks ago and my initial thought was this was a bad idea. (And judging by my voting, I didn't much care for it at some point in the past.) I have bunches of ancient questions on Stack Overflow that have been closed. In abstract, that's not a problem; I asked some broad discussion starters back in the day. But when I look through my closed questions, I get a pang of annoyance about some of them. How dare that moderator/the community close my perfect gem of a question!

So there are a lot of times when it seems helpful in the abstract to be notified of a closed question, but would only serve to make most people a tiny bit angry instead. However, Shog pointed out to me that close notifications might be very welcome if:

1. The question is closed as a duplicate of another question and
2. The asker's question hasn't been answered yet.

For duplicate questions, the close notification is functionally the same as an answer notification. In most cases, the asker will get a comment ping when a duplicate close vote comes in. (That has the pleasant side-effect of giving them a chance to self-close immediately.) But when the first vote is binding, no notice is sent. It does a disservice to the asker if we have essentially answered their question without notifying them. In this case, the notification should be something like:

duplicate
Grant the OP a binding close as duplicate vote
This question already has an answer here: New UI encourages askers to confirm or dispute duplicate votes

That way, the author can see their post if they like, but there's a link to the duplicate target which must be answered. The copy is taken from the banner on the closed post. That sorta mirrors the fact that answer notifications include a snippet of the answer.

But what about other closed questions? In my view, other close reasons are more likely to be confrontational. But we also know from the data that when people do edit their closed question, they can be reopened:

Close reason                   Edited %  Edited and Reopened %
------------                   --------  ---------------------

"why isn't this code working?"  19.73 %                13.14 %
recommend or find . . .          9.89 %                 5.29 %
duplicate                       38.47 %                 4.63 %
unclear what you're asking      11.37 %                 9.22 %
too broad                       20.48 %                 7.14 %

(These numbers come from this report. I've only included close reasons that make up 5% or more of all closures. The second percentage indicates how effective the edits were toward getting the question reopened.)

It might not seem like it, but those are encouraging numbers. Letting people know about closed questions so they can edit them might allow more questions to be rescued. In particular, questions that go unanswered because they are unclear or don't have enough information do get reopened after an edit at least some of the time. There's often some discussion in comments about the reasons for closing a question, but responding to those is a waste of time. Instead, any close notification should link to the edit window:

closed as off-topic
What are Stack Overflow unicoins?

For non-duplicates, close notifications are most useful if:

1. The question has not yet been answered and
2. It's not been a long time since the question was asked.

Again, I don't care to know if my question from last year was eventually closed; I've moved on. But it would be nice to know if my unanswered question from last week could be reopened with a strategic edit.