There was an article on HN a while ago from someone whom decided to start signing his reviews. To me, that seemed like a decent initiative.
The only real downside I see is that it allows the authors to contact and potentially bias the reviewer to affect a potential review of the written material.
Well, the other downside is that reviewers may be unduly lax in signed reviews. After all, people are often emotionally invested in their work / publications, and rejection sometimes offends.
SciPy did signed reviews for the 2017 conference the opposite reason. Obviously SciPy is not Science or Nature, but still. They cite the idea that if your review is tied for your name, you probably aren't going to do a crappy review because it will hurt your reputation. There's probably a balance.
I'd expect that people might soften the wording of their reviews, but to the detriment of their own time (i.e. spend more time by offering the same critique more constructively). I don't worry that people might soften their reviews by simply being less critical.
The only real downside I see is that it allows the authors to contact and potentially bias the reviewer to affect a potential review of the written material.