It is too easy to see the priests and teachers of Alma's people in the same light as we do our modern priesthood offices. When the verse notes that the "did watch over their people…" we can easily see in that phrase the home teaching (and visiting teaching) programs of the modern church.
While this would be correct for the modern church, for Alma's people this is much more literal. These priests and teachers were the government, and as the ruling organization, were very literally watching over the people. There is no indication of any other type of political organization, and so all aspects of community life are being directed by the body of men who are considered first religious and only secondarily political.