Flight instructor reported while departing a non-towered airport failing to monitor the CTAF for traffic information resulting in entering the runway while there was another aircraft taking off in the opposite direction. The other aircraft rejected their takeoff and exited the runway.

Narrative

Runway Incursion At Uncontrolled Airport Landed at ZZZ on an instrument cross country training flight. Winds were calm; and traffic was using runway XX. I had seen that the airport was empty once we had landed. We alerted the ctaf that we were going to taxi back to runway XX; and there was a runup area where we had parked to obtain our IFR clearance. We had some confusion contacting the clearance delivery and were not able to receive clear communication which resulted in us getting a clearance from the phone number which seemed very different compared to what was originally given. At this point I was told by the controller on the phone to go on ahead and contact the clearance delivery frequency again for the IFR release time. At this point I had my student start taxiing to the hold short line while I called up clearance delivery again. We had not made a call to the ctaf due to being right next to the hold short line and not seeing anybody around for runway XX. My full intention was to contact the ctaf after receiving the ifr release and then alert the airport that we were holding short and what our intentions were; followed by the proper before takeoff flow and checklist. However upon contacting ifr clearance delivery; after being given a. 4 minute release time; I quickly double checked with them what transition they wanted us to do for the ZZZZZ departure seeing as how we were given a clearance that did not specify. The controller then said that we were supposed to be doing a transition that had not being given to us in the original clearance from the phone number. This then led me to state that we were fine with letting the ifr release time expire while we figured this out because we had two different routes and a miscommunication. The approach controller then said to wait for a bit while they figure it out. However they quickly came back on frequency and said that the clearance you have is good enough and that our void time was now 3 minutes. As the CFI I know not to fall victim to being rushed when I'm not feeling ready; but I let myself anyway. I felt the urgency in their voice to get off the ground and so I quickly switched to the ctaf; configured the plane quickly for takeoff; and alerted the ctaf I was taking runway XX; thinking that the airport was empty. However; little did I know that there was actually a plane on the opposite runway taking off. I assume that they were making calls to the ctaf which I did not hear because I was on the clearance delivery frequency for too long. The other plane then said on the radio that they were rejecting takeoff for runway XY as we began our takeoff roll. In shock; I quickly responded on the comms preparing to reject takeoff; but then the other aircraft said it's alright just takeoff we already exited. So we kept rolling and departed. This was an absolute lack of responsibility on my part as the CFI. I had folded to external pressures and let that distract me from the procedures that ensure safety. I should have trusted my instincts and not taken off in a hurry if I was inadequately prepared.