If you have a weather radio that has SAME decoding and you have it properly programmed for your area of interest, usually your county, then it will only alert on weather watches and warnings that affect your area.
See this page on the National Weather Service's web site for more information:
NOAA Weather Radio
Most of the receivers listed by the NWS will have to be ordered from somewhere. The exception is the Midland WR-100 which has been commonly available at major retailers and has been promoted by some local TV stations. However, it does have an issue properly decoding the "end of message" signal from NWS. Supposedly, its replacement, the WR-120, addresses this problem. It should now be available at Best Buy, Walmart, etc.
Several of the Uniden scanners have SAME decoding. They have to be in weather alert mode in order to get the alerts which means you can't leave them in normal scanning mode and expect to get the weather alerts.
A fairly recent entry to this market is the Sangean CL-100 which is an AM/FM clock/radio with weather alert. I have two of these and am satisfied with their performance.