I don't think there are any city/town owned rinks, at least in the American West.
I can speak for the state of Arizona only on this, but Jay Lively Activity Center in Flagstaff as well as the Tucson Convention Center in... well, Tucson, are city-owned, but the TCC is operated by a private company (I couldn't find any information on the operation of Jay Lively). All six rinks in the Phoenix area are private. The city of Glendale owns Gila River Arena, but that doesn't really count.