ZSE Controller providing OJT described an airspace event when non RADAR routing procedures between ZOA and ZSE were not followed and an aircraft flight status was confused; the reporter alleging LOA compliance and timely coordination would have prevented the mishap.

Narrative

I was the instructor on R10; watching a student training on his last R-Side Sector. Three low altitude sectors where combined; very light traffic. Aircraft X called on frequency 'Seattle Center; Aircraft X with you.' The student told the aircraft to say his position; altitude; type aircraft and destination; assuming he was a VFR aircraft requesting flight following. Aircraft X came back and gave his position 200 miles South of Redmond at 100. Not the full answer we were expecting. Student then asked him to say his request. Aircraft X said he was IFR to RDM. I prompted the student to ask him if he was IFR or requesting an IFR clearance. Aircraft X then said he was on an IFR clearance and lost communications with ZOA. We verified his position; but still didn't have him on RADAR. He was in an area of poor RADAR coverage for both ZSE and ZOA; with poor radio coverage with ZOA. We had a D-Side working with us. They immediately called ZOA 43 to see what they knew about this aircraft. They informed the D-Side that the aircraft was not in our airspace. Our D-Side and ZOA 43 went back and forth on a number of issues. We were able to identify the aircraft which was in our airspace by then. Aircraft X should have been put on a Non-RADAR routing along his filed route of flight per LOA. Recommendation; have ZOA comply with the Non-RADAR routing in the LOA. Coordinate earlier with a data block only transfer.