Quoting dsmfever:
what if they decide to not sign it over?
Then you'll have learned what numerous people on this board alone keep saying, "Don't buy a car from ANYONE if the title is not DIRECTLY in that persons name".
Multiple sign overs, at a bare minimum, will require huge penalties for late/lack of official transfer. Car could easily have been stolen from the Titled owner for all you know. And yes no matter what state you are in a Lost Title search WILL require posting the VIN nationally and pinging the last registered owner. And yes it can lead to a visit by Johnny Law and special bracelets, least case loss of the vehicle.
Titles are the ONLY proof of ownership you have for that vehicle. The only signature for signover that matters is the one on the front unless it's from a registered Auto dealer ( altho it won't leave the dealers lot without your name on the title for processing ).
Man up and start the process in WA via electronic communication, hope for the best but don't be shocked if it goes badly.