Corporate jet pilot and helicopter pilot each reported a near miss after takeoff due to a communication error and a course deviation after takeoff. The corporate pilot identified the helicopter and maneuvered to avoid a collision.

Reporter 1

I was in Aircraft X given takeoff clearance on Runway 05 from Naples Tower with a right turn heading 060° to maintain 2,000 ft. I was entering the runway when Aircraft Y (with a request to fly along the shoreline) was given takeoff clearance at their own risk but given instructions to fly south - however they were not explicitly told I was taking off. I was on the takeoff roll after my abort speed about to rotate when I saw Aircraft Y cross the runway approximately 3/4 down the runway. I was accelerating through V2 and my flaps were at 15° so I knew I had enough speed to climb more aggressively to clear Aircraft Y vertically rather than bank to avoid it. I called on the radio and said something to the effect of "did Aircraft Y really fly across Runway 5?" Tower came back, sounded surprised, and said Aicraft Y was instructed to fly south and Aircraft Y said "I'm sorry I thought I was instructed to fly south once I got to the shoreline" which was absolutely not correct. Aircraft Y lacked situational awareness flying across the active runway in use (Aircraft Z took off just before us), didn't listen to ATC instructions, and ATC didn't catch the error until after I said something on the radio.

Reporter 2

I was looking for a departure from the helicopter landing area to the coast to proceed southbound. Although I can't recall exactly what I clearance I requested, I was not clear enough and I believe the Tower Controller understood that I wanted to proceed southbound from the parking area. I thought that I was given clearance to the coastline but I suspect this was my expectation bias and that the controller cleared me to proceed south. I am clear that I did not receive a clearance to cross the active runway and this should have caused me to stop and re-confirm the clearance. I climbed ahead and turned towards the coastline crossing the active Runway 05 in the process and an aircraft on the departure end of 05 reported that I had passed in front of him. The controller instructed me to turn to the north immediately and I complied. The controller then cleared me to turn southbound along the shoreline which I did and he gave me a number to call for a possible pilot deviation. I believe a contributing factor was my lack of sleep (approximately 3 hours) the prior night as I was concerned about the weather for this flight (as it had been poor the previous day). This should have caused me to be extra cautious with clearances and other complex tasks or even postpone the flight to the following day.