Around here, they're pests, non-native intruders on the territory of native beavers. They are legal to hunt year-round here in Oregon. (Along with Badger, Coyote, Opossum, Porcupine, Spotted Skunk, Striped Skunk and Weasel.)
I live inside the city limits of Portland, where it is illegal to discharge a firearm in the city limits. But I'm lucky to live in an area with quite a few "green spaces", including one that runs through my (very large) back yard. We have coyotes and the occasional nutria that run through the green space. My next door neighbor has been known to use rat shot or low-powder loads from his silenced .22 to help take out the pests. I asked our neighborhood resident police officer about this once, and he gave me the required "if you ever catch someone discharging a firearm in the city limits, call 911 immediately" followed by a wink-wink "but don't expect a response if he's safely taking out coyote or other pests." AKA: Go ahead.
I wouldn't trust ALL Portland police officers to think that way, though. (Heck, knowing Portland Police reputation for being thugs, my neighbor is very likely the exception, not the rule.)