I live in WA and have since 1970. I have carried since 1970, and I have kept up on the laws..
IN WA> you do not have to do anything. Private sale is just that, Private...rifle or pistol...as long as you follow federal law (resident of same state, and not a prohibited person)
You can ask the person if they are a prohibited person, and if they are a resident of WA,,,or not. You can ask for proof,,,or not...the requirement is that you "not Knowingly" sell to someone that is prohibited and is not a resident of the state.
If you Knowingly sell to a prohibited person...that is where you get in trouble.
IMHO: Ask them...are you resident of WA and secondly, are a person that is prohibited, by federal law, from owning a firearm. If they say they meet the qualifications, and you agree on a price...that should be all you need.
Some people want proof...my thought on that is...a person is innocent until proven guilty...NOT guilty until they prove themselves innocent.