Well, I can only speak from my previous experience.
There was a small group who owned the site (Quarter million members. Not a small site.). There were the administrators. They met together and made final decisions on major issues including who got to be a moderator. Often times, an existing moderator would notice that someone seemed very active in a particular sub-forum. They might note said person would flag problems, and was always very reasonable in their explanations of why something should be moderated. That existing Moderator would mention to the admin team that they thought such-and-such member would be a good moderator. From there, the administrators would talk it over, review that person’s history, etc. and make a decision.
Once approved, an invitation was sent to such-and-such member asking if they wanted to be a mod. If they agreed they were in.
After that there was a training period and then you would be introduced to the forum members and go live.
Moderators were autonomous within their assigned sub-forum/fora, but we worked together on big issues. And, we helped out if things went pear-shaped while the assigned mod was offline. We always consulted on things like whether to recommend someone for banning, and when a thread, post, or person was borderline, we would discuss what action to take. As a team, we worked very well together.