I have bought out of state trailers a few times, so here goes from memory.
1) You need a signed title for the trailer. If you do not have a title, you will need to run the trailer through the inspection process.
2) You need a signed bill of sale with the sales price listed. This is so WA can charge the sales tax.
3) They also need to have a weight certificate on the trailer. If the title shows the weight, you are home free. If there is no weight on the title, then you need to take the empty trailer to a certified scale (truck stop) and get a weight certificate. They are looking for empty weight, and they charge a fee based on the weight.
If it is a professionally built trailer (TPD, Interstate, Wells Cargo, etc), it is easier to register. If it is home built and never registered in WA, you have a 50/50 chance they will need to inspect it for a VIN number or assign a VIN number.
Mike