I think a first step is to make reviews public, in particular also rejects. In particular the big journals like science and nature seem to rely on a very small set of reviewers and often papers get reviewed by nonexperts, we had a paper rejected when a reviewer contradicted scientific fact that is in textbooks. When we protested the decision they just send it back to the same reviewer who stayed with his claim. I think if these reviews would be public the journals would get much clearer feedback on how bad some of the reviewers are.
> I think a first step is to make reviews public, in particular also rejects.
As a reviewer, I have no objections towards my reviews going public. However, I have often had to tell authors that their paper is dumb (not in those terms, obviously), and I'd rather keep those reviews private. Being rejected is hard enough without the entire world knowing how much your paper sucks.
Of course, there are terrible reviewers too. But IMHO that's better solved via better area chairs than with a Twitter mob. I would rather not being 4chan into the review process.