Yes, other states must abide by WA laws at time of purchase in those states, and vice versa, per federal law.
Here's the exact interpretation for rifles (handguns you simply just can't buy out of state):
"Under federal law, individuals can lawfully obtain a long gun from a seller or transferor in another state, provided that:
Both parties meet in person to conduct the sale or transfer.
The sale or transfer is conducted by, or through, a licensed dealer, pursuant to a background check and other requirements.
The sale, delivery, and receipt of the long gun fully comply with state law in both parties’ states of residence."
That was even before I1639. Granted, stipulations in that new law make it rather difficult for an out of state Sportsmans store to follow the WA law for the specific background check for WA, so yes as carpsniperg2 points out, has to be via WA FFL.
Just waiting for the lawsuit challenging I1639 to work it's way through to the US Supreme Court. The ban on selling a gun to an out of state resident seems like the part of the law that should at least get overturned, as that falls under interstate commerce laws as well (ie an arbitrary restriction of trade).
