With the latest server update, town permissions have changed a bit. By default, town citizens are now only able to pick up their own blocks (though they can still place them anywhere)
If a mayor trusts a citizen a great deal, they can promote them to a Town Official. Officials have permission to take ANY block from the town. Mayors should be very careful who they add as an official.
To designate a town citizen as an official, the mayor can type /townoff <player name> in the chat window.
Thanks to everyone who submitted ideas about this!