FLC OF LTT MISTAKENLY TOOK OFF WHEN NOT CLRED RESULTING IN AN AIRBORNE CONFLICT.

Narrative

WE WERE HOLDING SHORT OF RWY 1 WAITING FOR A KING AIR TO LAND. AFTER THE KING AIR LANDED WE WERE TOLD TO TAXI INTO POS AND HOLD AS THE KING AIR WAS TURNING OFF THE RWY; WE HEARD INSTRUCTIONS TO 'TURN L TO 270 DEGS; CLRED FOR TKOF RWY 1.' BUT DID NOT HEAR AN AIRPLANE IDENT. SINCE WE WERE HOLDING ON THE RWY; WE ASSUMED IT WAS US THAT THE TWR WAS TALKING TO. I REPEATED BACK THE INSTRUCTIONS TO THE TWR AND WE STARTED OUR ROLL. AT ABOUT VR; WE NOTICED A CESSNA 172 AHEAD AND BEYOND THE END OF THE RWY LIFTING OFF AND CLBING OUT OFF THE SAME RWY. THE TWR INSTRUCTED HIM TO TURN L TO 270 DEGS SO WE QUESTIONED OUR HDG OF 270 DEGS; WHICH THE TWR TOLD US TO FLY RWY HDG. THE RWY WAS 10000 FT IN LENGTH AND THE CESSNA HAD DEPARTED THE LAST 4000 FT. WE WERE UNABLE TO SEE THE CESSNA DUE TO THE CURVATURE OF THE RWY; AND ASSUMED THE TWR WAS TALKING TO US SINCE WE COULD NOT SEE THE CESSNA FROM THE BEGINNING OF RWY 1. EVEN THOUGH IT WAS OBVIOUS TO US THAT THE TWR WAS TALKING TO US; I SHOULD HAVE QUESTIONED THE TWR WHEN I DIDN'T HEAR AN ACFT IDENT.