C182 Flight Instructor reported on take off observing unannounced traffic climbing from crossing runway; resulted in taking evasive action to avoid NMAC.

Narrative

Reporter pilot operates out of ZZZ often; about 3 times/week. Came in to refuel. When we landed and subsequently took off; winds were 040/8-10 kts. as reported by AWOS. We chose to land Runway XX; since it is a much longer runway and other people were in the pattern for that runway. After we refueled and were almost finished conducting our run-up; a float plane taxied to Runway YY for takeoff. No other planes were speaking on the frequency. I heard a call from a jet that was 12 miles out on an approach to Runway XX. At that time; I saw a plane that either could have been on a weird downwind for Runway XX or possibly leaving the area; so after the jet called his 12 miles out; I called and asked if there was anyone else in the pattern for Runway XX (not a call I usually make; but since I could not see where that plane went due to the position of my plane by Runway YY and the trees that block the view there; I decided to make the call). In response to my question; the jet repeated his 12 mile out position. I thought possibly the call of another plane was blocked by the jet; so I said; 'is there anyone else in the pattern for Runway XX at ZZZ besides the jet'; to which there was no response. As a government agency; we operate the Cessna 182 as a crew. We conducted our pre-takeoff checklist and made a call that we were departing from Runway YY. Still no other calls. I was acting as flight instructor at this time; because the copilot I was with does not fly frequently. I was not the pilot flying. As we were rolling and off the ground; I looked to the right to check Runway XX again and there was an aircraft that had either done a go around; or a touch and go and was also airborne; but from the intersecting runway. We were on a collision course and because I could not get the words out fast enough; I said my airplane (prebriefed; so the pilot flying relinquished controls) and I pushed forward on the yolk; as that plane was closer to being above me than below me (they may not have seen me at that point; I'm unsure because their nose was up and we were at their 12 o'clock position.) After passing by the aircraft; I reinitiated a climb and simultaneously made a radio call saying that we just took off from Runway YY and have the aircraft that just departed XX in sight and asking if they are on the frequency. The aircraft did not respond. I watched the aircraft turn crosswind and downwind without making any calls as well. I called them out for the jet; who was probably 4-6 miles out at this time (guessing). An observer on the ground told me the next day that he thought he was about to witness a collision. He also said that long after we left the frequency the aircraft starting making traffic calls.I do not recall our TCAS making a 'traffic' call; but it could have. We do not have ADSB out as we operate with a waiver.