- most academics I know are paid for reviewing (via their employment contract)
- thus it would be up to the employer (university/research institute) to charge for the service
- I'll outline the burocratic and legal steps needed on the reviewer side (in my country: Germany) to be able to charge personally for a review. For many academics, this burocratic offset compared to what you can earn this way is not attractive.
First of all let me say that from my experience
academics do pretty much all the work for free
I think this is true at best for a very small minority of academics. My work contract says that (among other things) I'm paid for "publication activities" and that clearly includes reviews. Clearly as in administration asks me to report for yearly statistics number of reviews done for which journals just like they ask for manuscripts, oral presentations and posters.
So while I'm not paid by the publisher, I am paid by my employer for the reviews. And I know very few academics who go on publishing and reviewing after the academic job ends - few people put that much effort into a hobby.
(I'm thinking here more of graduates/post-docs without job than of retired professors because I think the out-of-job-academics are the better control group for this discussion)
For me this makes the question very similar to why does a car mechanic work who is employed by a workshop not charge the customer directly? Answer: it's the employer who charges the customer (via their administration), the mechanic is paid by their wage.
We now may ask why doesn't the employer (research institute, university) charge the publisher for services received? IMHO this is a sensible question and one that actually should be asked. Edit: However, to me this is not the same as the question asked why a single researcher doesn't charge the publisher. The standing and the aims in these negotiations are IMHO totally different of a single researcher compared to a university/research institute.
[slightly off topic: one answer to this may be that for academic institutions of a certain size the number of reviews done by the staff comes close enough to the number of reviews needed for the publications of the same staff - so introducing payment for reviews (including the institution needs to pay for the reviews they receive) just means that more VAT needs to be paid, and thus generates a net loss.]
Edit: Why do I think that charging for review will lead to charges for having your paper reviewed? For one thing, of course commercial publishers won't like to diminish their profits if they can avoid it. But even then: assuming an open source source publication fee of, say, 1500 EUR/US$ leads to 500 EUR/US$ profit (that's 33 %) for the publisher means that 500 EUR/US$ could theoretically be spent for the review before the publisher will enter the loss zone. That pays (see below) for maybe 5 - 10 h of professional academic review time. I often spend considerably more on a single review (see e.g. https://academia.stackexchange.com/a/5799/725 where @JeffE cites a rule of thumb of about 1 h/manuscript page - we often have 15 - 25 manuscript pages). I don't know what the average number of reviews is per published paper, but I'd guess that it is somewhere near 10. But even a low guesstimate of 3 for the first round + 2 for the second (that doesn't even include that papers are declined!) means that 5 x 20 h = 100 h of review time per published paper. That's something like 5000 - 10000 EUR/US$, or 10 - 20x the huge profit of the publisher.
So, yes, the reviewing is a huge amount of work and it needs to be compensated, like writing the papers needs to be compensated. And yes, it is the researchers who do this. But even the most evil shark publishers with 40 % profit of a 1500 EUR fee for an open access publication would be able to pay for that.
How to charge a publisher for services received (i.e. the review) as the reviewer in person?
(this is for Germany, other legislations will differ)
On the other hand, well, yes: why not charge the publisher?
Scientific reviewieng is a classic professional service of freelancers (German: freiberufliche Tätigkeit). In order to do such freelancing, you first of all need to make sure that there is no conflict with any employment. This is best done by exchanging a couple of letters with your professor/director/administration. Unless they actually have some deal with the publisher, they probably wish you good luck and are happy with that.
You then go to the tax office and ask for a freelancing tax number. If this is just for the few reviews you do, you'd also ask for exemption of VAT, otherwise you'd need to do VAT declarations (if dealing with a publisher in a EU foreign country, things are more complicated). But at the very least, the freelancing tax number means that your income tax declaration becomes mandatory, gets an additional set of forms for the freelancing and you have a shorter deadline to hand it in.
You'll need to put in this effort regardless of whether you are actually able to convince a publisher to pay you or not.
I guess most researchers I know think that compared to doing the review for the academic salary this extra money is not worth the extra burocracy. (I know of one colleague who told of his burocratic experience where an extra payment < 1000 EUR for a seminar was concerned)
Now, I anyways do freelancing, so the marginal effort for me is low (as it is for people who anyways do vocational tax declaration and do it early). However, so far I have to say that the market price of reviewing is rather unattractive. I've been offered 100 US$ to do a quick "review" within a couple of days (no full review was asked but rather something like an opinion on the manuscript and pointing out which points I'd recommend should be addressed before actual submission).
I estimate that I cost my employer about 75 - 100 US$ / h (of which 22 US$ / h actually arrive at my bank account after taxes + social insurance for me have been paid). Thus, the offered market price for the review service boils down to the cost of 1 - 1:10 h for my employer. Even though my freelancing doesn't have as much overhead (e.g. because small-scale freelancing on the side is covered by the social insurance of the employment contract), I'd need to finish that review within ca. 2 h to get the same hourly wage I get for my employment contract. Also as it is very much on-demand of the journal, it cannot even be used too well for filling up time when I don't have a customer. So all in all, even though my marginal burocracy for doing this is low, it is not super-attractive. In fact, I'm better off if I can put the time onto my academic time sheet. You may decide differently, particularly if your employer does not accept a time sheet.
Another comparison: for me, the hourly wage for such a review is close to filing VG Wort claims - though the hourly wage there would be better if I had more papers to file, as there's quite an offset of remembering how to do things that need to be done just once per year.