A Tower Local Controller reported their trainee cleared an aircraft for takeoff at the same time a taxiing aircraft had been cleared to cross the runway.

Narrative

RyXXR was active; and I had just taken the LC (Local Control) position to conduct OJT. The GC position had just been relieved as well. Aircraft X called for departure from RyXY. The trainee requested an IFR release for the RyXY departure and coordinated control of the alternate runway. Aircraft X was cleared for takeoff from RyXY. This was the only aircraft under LC control at the time.I saw Aircraft Y taxiing from their normal parking place; which is very close to the runway. I saw Aircraft Y taxi up to the hold short line for RyXY and immediately got the feeling they were not going to stop. I saw Aircraft Y cross the hold short line. I yelled to get GC's attention; as they seemed to be giving clearance to another aircraft. At this time; the CIC (Controller in Charge) jumped up as well; and tried to get the GC controller to stop the aircraft. I keyed up to tell Aircraft Y to stop; before remembering they were not on my frequency. In those 2-3 seconds; Aircraft X was still on takeoff roll at full speed. I recognized that Aircraft X was at or near rotation speed; and I felt that telling them to abort takeoff would be more hazardous. With Aircraft Y crossing the middle of the runway; and Aircraft X ~1000 ft away; Aircraft X rotated and climbed over Aircraft Y.The pilot of Aircraft X responded 'That wasn't good' as he departed. We apologized; and switched them to departure. Aircraft Y also asked if they had made a mistake. Being a local pilot; I advised them that if we needed; we could get in touch with them; but we were looking into it. The GC controller recalled that he wasn't sure if he told Aircraft Y to hold short; and felt that he would have remembered if he did. After listening to the recording; we confirmed that GC told Aircraft Y to cross RyXY after the runway had already been transferred to LC. The memory aid for runway control was used properly to reflect this.We have a Runway Incursion Device that indicates 'ownership' of the runways. It was being used and correctly indicated the runways owned by LC at the time. Memory aids only work if we remember to pay attention to them. Additionally; Aircraft Y was following the instructions he was given; but it is prudent for pilots to verify a runway is clear before entering the runway. I have seen multiple instances where planes even identified other aircraft landing or taking off; asked about the aircraft over the radio; and continued to try to cross.