A related question is "who pays the per-paper fee". Well, not me, and none of my colleagues. However, I've learned that the fees do get paid, but not by individuals. Rather, it's libraries that are paying, to fulfil interlibrary loan requests for papers in journals with high subscription costs and low readerships. If the number of requests times the price per request is lower than the yearly subscription cost, this can be the only way to stay within budget.
Where I work, research grants let me cover journal "page charges", but they do not let me subscribe to journals, nor contribute to the library budget. This has always struck me as crazy, for the yearly cost of many journals is under the cost of publishing a single paper. Heck, we have researchers publishing in the journal of whatever, paying several tens of thousands of dollars per year ... and our library cannot afford to subscribe, so we can see our work.
Yes, you read that right: in many cases, authors have to pay to publish, and then pay again to read.
Some progress is being made, though. Many granting agencies now encourage researchers to publish in open-access mode. That would seem great, except that they did not increase grants to compensate for the extra cost. Money spent on those fees is money that cannot go to support graduate students.
As others have noted (here, and in dozens or hundreds of similar discussion threads), things are different from field to field. I work in a field where it's just as important to read old papers as new ones. That means that the solution of authors posting to their websites is not always viable. Probably a third of the papers I read (in support of teaching) are from people who died long before the web got started.