I'm still reading the same issue of Nature (I do do more than read journals at work), and another editorial is on the topic of peer review.
Interesting points it raises, and I have to say I haven't had any formal training on the topic. I do get asked to review papers - I suspect the standard of training is that a new scientist gets picked to peer review papers by the editor of a lower impact journal who just wants someone to cast an eye over a badly written manuscript and declare it really needs major rewriting to become comprehensible.
This is largely what I get, to be honest - articles that can't claim the excuse that the author's first language isn't English (although that doesn't help some of the tortured grammar), as the article as a whole hasn't been thought out. Too often I get a generally vague list of results with a conclusion at the end of it that may only relate to some of them. It gives the distinct suspicion that there are a lot of labs out there where the PhD student is expected to publish, and the professor appears to be expecting that the peer reviewer will do his work for him and go through the paper to correct all the mistakes.