Flight Instructor reported a critical ground conflict when another aircraft took off from the opposite end of the runway at the same moment.

Narrative

I was flying Aircraft X. I was getting my night landings in to maintain my currency and to build up my nighttime for ATP mins. The pattern for the past 2 hours was for runway 13 at Falcon Field (FFC). During that time I was performing full stop landings and taxiing back to runway 13 on taxiway A. I had also communicated with another aircraft as they did a full stop landing on runway 13. With a regular radio check and the fact that Aircraft Y responded to me; I knew my radio was fully functional. There was an aircraft that had called a 6 mile final for runway 31. I have a G430 with WAAS capability and it also has traffic. I did not see the other aircraft on a 6 mile final. I called on the radio to clarify that the aircraft was on a 6 mile final and got no response. During this time; I was on taxiway A around the runup area on the northern end of the field. I saw aircraft lights on a base to final turn for runway 31 and announced on the CTAF that I will hold short of runway 13 for the traffic landing on 31. The winds up at pattern altitude that night were favoring 13. They had announced that they were landing for a full stop and did not say whether they would be taxiing back to runway 31 for an additional takeoff. After they landed; I saw that they were well clear of the runway and had announced that I would be departing runway 13 with a right closed pattern. I had taken the runway and was already at full power; and at 55 knots for rotation speed. I heard them say that they are departing runway 31 and I saw the aircraft race onto the runway with no hesitation. During this time I was carrying too much energy to abort the takeoff safely and avoid the other aircraft; so I continued my climb off the ground. I was on the radio trying to communicate to the other aircraft that I was taking off and I had to sidestep to the right (there was a crane on the left). They finally acknowledged me on the radio after I was constantly trying to get them to respond. I did not get their full tail number; as they were only saying the last 2 letters. At this point; I was above the pattern altitude at 2;000 feet indicated and saw that they continued with their takeoff and flying a left pattern for 31. They called and asked if I was the aircraft on final for 13 that they almost had a head on collision with. I had pointed out that I had the runway well before they took runway 31 and they said; 'hey we're really sorry about that'. In my frustration; I said to them that they need to pay attention. I had exited the pattern and set up for a 45 to the downwind for runway 31 to avoid any more conflict. They did not announce crosswind; just that they were downwind and were a full stop landing. After they had landed; I spotted where they had parked their plane and made a full stop landing on 31.