I have looked into this question a couple times since Publons became active. There does not seem to be any other platform with the same functionality available. So I use Publons, albeit in a very minor way.
The Publons site let's you take credit for reviews, but as part of prepublication refereeing and post publication reviewing. You have the option to make as much information about the reviews public as you want, including the full text. I don't really use it for anything except as a service to validate that I have completed as many reviews as I say I have.
Every time I get an e-mail acknowledgement of a referee report I have submitted, I just forward it to firstname.lastname@example.org. Coming from the e-mail address I have registered with Publons, the site knows that these are referee reports to be added to my profile. For journals that are hooked into the Publons system, the processing is very fast; for other journals, it takes a few days. And no information about the confirmed reviews appears on my public Publons profit, except the date and the journal involved. I could make more visible, but it defaulted to that minimal information.
When I submit my annual statement of activities, I thus have an external site to attest to all my reviewing work. In fact, I've never needed it; my self reporting had never been doubted. But if it were, Publons would be there to back me up.