For example, SlashDot assigns random users moderatorship powers for one story. This makes the moderating a reflection of the average Slashdot reader's opinions, on average.
Actually, Slashdot uses readers' karma (their "ratings" essentially) to weight the choices, so the choice of moderator isn't so random and can lead to a feedback loop where a cabal of high karma users keep donning karma upon each other. (You cannot give karma to yourself.) This is probably a bad thing.
Also, randomly selected individuals from a given population will tend to have roughly average abilities. This will tend to pull the output of a RandomPeerPrivilege systems towards an "average" level.
However, there is some level of self-selection: not everyone granted moderation points will use them.
One major problem with slashdot's moderation system is that moderators view the same already moderated view of the posts that other posters do. Thus, early-moderated posts to a story tend to get moderated up repeatedly, while late posts tend not to be seen by moderators at all. Random moderators would have to view random samples of posts to make the system give an equal chance of moderation to every post.