General aviation instructor pilot reported a near miss with another aircraft while departing a non-towered airport on a training flight. The instructor misidentified a white light on the far end of the takeoff runway as a vehicle; but was an aircraft planning to takeoff opposite direction. The other aircraft heard the instructor's communication and exited the runway; as the instructor departed.

Narrative

During a dawn depature from FLV I had a near miss with another departing aircraft. I was conducting a training (second flight overall) with a student. I was on taxiway Alpha departing RWY16 about 3 mins after another aircraft. I had all my lights on (strobes; beacon; and landing); and made a CTAF call. When clearing the runway I noticed a white light on the far end of the runway; which looked like another aircraft. Howver; it was moving laterally and white; so I assumed it was a ground vehicle making a circuit of the airport perimeter (this always occurs at Leavenworth as people conduct physical training or hiking on the airport perimeter road). Moreover; another aircraft had just departed 16; so I assumed that no aircraft would immediately takeoff 34.I heard no radio calls on the CTAF as we entered the runway. We tookoff normally; lifting off about 1500 feet before the departure end of the the runway. On the downwind I noticed traffic departing 34 (opposite direction) both visually and via ADS-B. I again heard no calls. The departing aircraft turned northeast. I immediately checked my radios and found the student (or me) had turned down the #1 radio; which had been on CTAF. However; I still heard the previously departing aircraft north of the field. He called me and stated that the aircraft which departed RWY 34 was trying to get a hold of me on CTAF (the calls I evidently did not hear). I tried to make an apology call on CTAF but again did not hear the aircraft. According to the aircraft I could hear; the third aircraft had heard my depature call and exited the runway (kudos to him).