Some do, but most are volunteers. I don’t know what the range is, but I suspect it varies a lot, depending on the income of the site. On Fluther, we find our volunteer moderators in our own community.
I never did. I think it’s rare for a mod to be paid solely for being a mod, though if being a moderator is merely one of the many duties of their job, things are a bit different. Regardless, I personally have never heard of someone getting paid to be just a mod.
One of the sites I moderate gives out gifts and has competitions that offer financial incentives, but I haven’t heard of a site paying a moderator. Some popular opinion sites might, especially if they are sponsored, such as FatWallet.com
Metafilter.com pays the mods a actual livable wage. I think they have six now. That is the only site I can think of that does. But Mathowie (the owner) is a Socialist.