PA23 pilot reported nose gear failure while landing and a runway excursion.

Narrative

We had just made a full stop landing at ZZZ uneventfully; and taxied back to the active for another lap in the pattern. We flew the pattern and brought the gear down; with three green lights indicating. On short final tower directed us to go around; so I added full power and brought the gear handle up and we proceeded directly back to ZZZ1. En route; my instructor noticed that the gear position lights were indicating that the two main landing gear were down; but not the nose gear; the nose gear light was not illuminated. This was unusual since we had seemingly retracted the gear on the go around at ZZZ. On downwind at ZZZ1; we lowered the landing gear handle and the nose gear light still did not illuminate. We broke off our approach to troubleshoot; and my instructor removed the nose gear indicator bulb and replaced it with the bulb that illuminates when all three landing gear are up. Upon replacement; all three landing gear down lights illuminated; indicating that all of the landing gear were down and locked. To be sure all the landing gear was down and locked; we flew a low pass over the runway so someone could inspect; and we were subsequently told over the radio that it looked like all three landing gear were down. The owner of the aircraft also verified to us over the radio that when we switched the bulbs out and all landing gear down lights were illuminated; the gear should be down and locked. With that verification; as well as being told that the landing gear visually looked down and locked; we came in to land. The owner also drove down to the landing end of the runway to visually inspect whether or not all the gear was down while we were on final; and he gave us no indication that anything was wrong until we had touched down on the main landing gear; at which time he said over radio; 'nose gear's not down'. By then; we were firmly on the ground and not able to react in time to the last second announcement. We subsequently veered off the left side of the runway where we came to a complete stop shortly thereafter.