PA-23 instructor reported a nose wheel landing gear malfunction while entering the traffic pattern at a non-towered airport. The instructor coordinated an observer on final approach to verify the gear was down but was not told the nose gear was not down until the aircraft was landing; and the aircraft veered off the runway with no injuries to the pilots.

Narrative

The purpose of the flight was a training flight. We flew the RNAV XXL into ZZZ1 airport; and both noted 'three green' multiple times prior to an uneventful landing. We executed a normal takeoff and flew the airplane in the pattern with the intention of landing OEI (One Engine Inoperative). We both noted 'three green' on final. ZZZ1 tower gave us a go around; and so we did. They gave us an early crosswind as there was a single engine Cessna ahead of us. We departed the pattern for ZZZ airport out of the downwind. While on a 3 mile 45 for the left downwind of Runway YY; I noted that the gear handle position was up; however two green lights were illuminated - the mains. As we were entering the 45; we dropped the gear to see if we could get three green; we did not. We performed a right 360 and attempted to troubleshoot. I unscrewed the amber 'gear up' light and switched it with the nose gear light to see if maybe the bulb was simply burned out. When I switched the lights and with the gear handle down; they worked - we had a green; an amber; and a green all illuminated in a row. We communicated this information to the aircraft owner over the radio. To confirm the gear was down; we coordinated with the aircraft owner and executed a low pass at ZZZ airport for a visual check of the gear. We received communication over the radio that the gear was down - the mains and the nosewheel. Given that information we made the decision to land the aircraft. We were told over the radio that the aircraft owner was going to drive down to the approach end of the runway to look as we landed. We did not hear anything over the radio on short final and proceeded to land. The mains touched down uneventfully; and at the same time the nosewheel came down we heard the aircraft owner say 'the nose gear's not down.' We veered to the left of the runway and came to a rest in a grassy area shortly thereafter. There were no injuries to the pilots or anyone on the ground. There was no smoke; fire; or any fluids leaking from the aircraft.