What are the benefits of serving as a reviewer, initially as a PhD/PostDoc and later as a faculty member/research scientist?
The primary advantage of peer reviewing is that you are ahead of the researchers in your field by a margin of several months. Second advantage of getting a bad paper for review is that you get an idea "what to avoid" in your own future work. Most people do not realize this because it is not a paid service, nor there is any public acknowledgement on the paper. Several journal editors have mentioned to me that they have difficulty inviting reviewers. I asked an editor of a pretty good review-type journal and he mentioned that at times he has to write to 10-12 people and only a few respond. It is a also common complaint among editors that people from certain countries don't commonly accept peer review requests because the academic promotion is based on paper publications rather than article reviewing.

In short, we should accept review requests, if we feel we are qualified to do so, as a service to the community as well for your own education.