Explanation of the purpose of this site
Here are four categories for questions of interest to moderators:
Questions about policy for the individual site: These belong oner-site metas
"What should we post on our blog?"
"Are questions about X on topic?"
Questions about policy on Stack Exchange in general: These belong on MSO.
"When should a question that's not really something the user faces be closed"
"What should I do with a user who constantly posts low quality, but not spammy, answers?"
Questions about specific moderation problems: For these, flag the user/answer and talk on per-site moderator chat rooms. They don't belong on MSO or Mods.SE because of privacy and temporality issues.
"This user keeps posting inappropriate comments"
"This question is under an edit war"
Questions about general moderation tools and abilities:
"How do you ban someone from both chat and the main site?"
"What happens when a mod flags a comment?"
"How do you ban a user who creates new accounts and posts spam from multiple IP addresses?"
The second category is a subset of Meta Stack Overflow. The fourth is what I want to see here, I'm not sure why MSO questions were posted and upvoted on this proposal, but the fourth category needs a home. That home is currently the Teachers' Lounge.
Everything about how to act as a moderator is an easy search through the chat logs and explained (quite verbosely now) in the tools themselves.
If searching chat logs is a better solution to Q&A than Stack Exchange, why do any of the other sites exist? IRC, mailing lists, or forums also have search features, but Stack Exchange offers something better that those sites don't have. Placing Q&A on Mods.SE would reduce the amount of time spent/wasted in the Teachers' Lounge and make educating new mods easier.
The tools themselves are running on live systems, with real people. Most of the tools are well documented, most work as expected, and most are reversible. However, not all are reversible, and not all are inconsequential. Discovery and learning by doing are difficult.
Reasons we need this site:
There are >500 sites in Area51 at the moment, which could develop into 1,500 to 3,000 moderators. That's not the global audience of the Internet, but it's an awful busy chat room. There's no reason not to plan for scale.
The sites are live. While you can go far playing on one's meta site, you need a sock puppet account to accomplish anything, and user, migratory, and permanent actions are hard to explore.
New mods don't search the chat room, because they don't know the right keywords to search for, there's a huge amount of noise, and it's inefficient and tedious. drachenstern estimated that 65% of the content of the Teacher's Lounge was just banter.