Reviewing a manuscript is tickling, but also heavy and responsible job. I'm unaware of anybody being paid for that. Some open access journals at most publish the reviewer's name. Those journals however press you to decide fast, e.g., within 10 days, with likely not best decision and motivations given. Some managers would even disregard your decision in order to publish the more they could and earn as much as possible. Thus the reviewer becomes just a marionette. "Good old" journals are more patient, but would not share looots of earnings with those who guarantee their reputation.
Compensating the reviewer would have a twofold objective. First, to guarantee an objective review, provided signed contract. Second, to compensate energy spent often outside of normal working hours. I realize that this might increase formalities, etc., by nowadays all those formalities could easily be resolved with the help of simple instruments, such as PayPal etc.
Even more sensible solution is to GIVE THE MONEY BACK TO THE UNIVERSITY (or researcher's employer). This would solve various formal issues (taxes, etc); would give more respect to the researcher in his institution (this could become a part of the researcher evaluation) and would give back lots of money given by universities, etc. to the publishers.
Last, note that respectable organizations, such as the European Commission, already pay project reviewers...
SO, are we ready to revolutionize the system? What actions are you ready to take? This ResearchGate platform might be a good tool for a successful initiative.