I'm nearing the 2,000 mark in reputation on Stack Overflow, which I believe will earn me the privilege to edit other people's posts. Is the "edit other people's posts" role applied at a site-wide level, or can it be granted for a specific set of tags in which I have demonstrated some level of competency?
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You get it for the whole site. Reputation tries to be a measure of an overall level of trust you should have, I think trying to dilute that down to tag-based permissions looks attractive at first but would end up being more trouble than its worth.
The rule applies to the entire site. The purpose for the threshhold isn't so much to prove that you are competent, but to require a not insignificant person investment in the site, which serves the purpose of cutting down on spam/abusive edits, as well as providing an incentive to post good questions and answers.
In other words, once you reach 2000 rep, the system trusts you to edit other peoples posts, but there's no implicit assumption that you are competent in that area.