How do you earn opportunites to review journals or conference papers?
Assume that I have just finished my postdoc and am now starting a professorship/scientist position (I am not). How do I start joining the ranks of peer reviewers? More importantly, how do I become a peer reviewer of a particularly prestigious journal rather than some junk journal? Does the journal contact me or do I start to publish in those journals, the editors recognize my "expertise" and then they solicit me? Even though everything is "blind" does my reputation of doing quality reviews get shared with other editors?
The best way to do it, in my experience, is to write. Every single journal I've been asked to review things for (with the exception of one done as a favor) was born of a request by an editor to review a paper in an area where I might be considered an "expert" - because I had a publication, and a decently received one, in that topic area.
Once you get an invitation or two, accept them and as importantly make your reviews useful and timely. Establishing yourself in an editor's mind as someone who gets their reviews back, and gets them back on time (or communicates when they aren't able to do that) can't do anything but help in ensuring that you'll get similar requests in the future.