The Second Amendment to the U. S. Constitution guarantees most people's right to bear arms. There are limits on that right, though, and it is the State the regulates the permits for concealed pistols, for example. http://www.dol.wa.gov/business/firearms/faconceal.html.
Part of the judgment and sentence for certain crimes can contain a prohibition against possessing firearms. RCW 9.41.045 (http://apps.leg.wa.gov/rcw/default.aspx?cite=9.41&full=true#9.41.045). Restoration of the right to bear arms can occur under RCW 9.41.047 (http://apps.leg.wa.gov/rcw/default.aspx?cite=9.41&full=true#9.41.047). Courts can order forfeiture of firearms. RCW 9.41.098 (http://apps.leg.wa.gov/RCW/default.aspx?cite=9.41.098).
So, it wouldn't be the federal law enforcement personnel who would ordinarily enforce any State court order disallowing possession of firearms, it would be State officers.
But, the Bureau of Alcohol, Firearms, and Tobacco can get into the act in some circumstances, too. (http://www.atf.gov/).
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The previous responder is actually incorrect on this issue.
18 U.S.C. s 922(g) indicates "It shall be unlawful for any person (1) who has been convicted in any court of, a crime punishable by imprisonment for a term exceeding one year;...to ship or transport in interstate or foreign commerce, or possess in or affecting commerce, any firearm or ammunition; or to receive any firearm or ammunition which has been shipped or transported in interstate or foreign commerce."
This is a Federal statute that creates a Federal crime. Read the rest of the statute for more details, but the short answer to your question is that if you have been convicted of a crime under Washington State law that is categorized as a felony (all felonies in Washington are punishable by imprisonment for a term exceeding one year), your firearm rights are restricted at both the State and Federal level.
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