PA28 instructor with student experiences conflict during landing roll with Piper Arrow attempting to land opposite direction. The Arrow pilot goes around. It is later determined that the Arrow pilot was not on the CTAF and was landing straight in.

Narrative

Arrived at SOW with student as pilot flying; I was PIC/Instructor. Arrived from SW; descended from 9500; heard a B1900 depart Runway 24; we asked their direction of flight & they responded right cross wind departure to the North. We advised area traffic of arrival from SW & proceeded to cross departure end of Runway 24 on cross wind leg then turned to down wind leg. We announced on crosswind & on down wind & wing-up turning base & 1/2 mile final (full stop) for Runway 24. We made normal landing & were about to turn off Runway 24 midfield when Cherokee Arrow observed approaching fast from Runway 06 departure - gear down. We expedited off runway & observed the Arrow pass overhead on departure from Runway 06 & raising landing gear - they appeared to depart on a left cross wind. We made a radio call in the blind to Aircraft departing Runway 06 SOW 'Do you have your radio on?'. We proceeded to transient parking & tied down; observed same Arrow land on Runway 06; turn off mid field & taxi to central ramp. Unicom operator informed us that Arrow had just established radio contact & acknowledged he was not on frequency. Unicom operator acknowledged Runway 24 is published calm wind runway; acknowledge they heard many calls from us. I was able to contact the pilot via telephone and he indicated he typically flies straight-in's to SOW since never any traffic; was listening & making calls from last 30 miles out; indicated he had left audio panel switched to AWOS - did not see/hear B1900 depart from 24 - did not hear us - did not know Runway 24 was designated calm wind runway. I mentioned that good idea to use lights within 5 miles of airport; AIM recommends strongly against straight-in's at uncontrolled fields; AFD lists Runway 24 as 'calm wind runway'. We were ALL very lucky that nothing bad happened here!