A BEECH F90 FLC TAKES OFF WITHOUT CLRNCAND CREATES A POTENTIAL CONFLICT SCENARIO FOR THE CTLR AT CAK; OH.

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TWR APPARENTLY CLRED US INTO POS AND HOLD RWY 23 AT CAK. BOTH THE CAPT (ACTING PIC) AND I THOUGHT WE HEARD 'CLRED FOR TKOF.' APPARENTLY; THERE WAS ANOTHER ACFT ON A PRACTICE ILS TO RWY 19 TO A LOW APCH. AS WE WERE ON OUR TKOF ROLL; I HEARD THE TWR CTLR COMMENT TO THE OTHER ACFT THAT HE APPEARED TO BE WELL E OF COURSE. THEN AS WE WERE ABOUT TO ROTATE; WE HEARD HIM TO THE EFFECT THAT WE HAD NOT BEEN CLRED FOR TKOF. IT WAS TOO LATE TO DO ANYTHING ABOUT IT; AS WE REACHED ROTATE SPD ABOUT THE TIME HE FINISHED THE SENTENCE. WE THEN HEARD HIM ISSUE THE OTHER ACFT A GAR WITH A CLBING TURN TO THE E. WE SPOKE TO ONE OF THE SUPVRS WHEN WE REACHED OUR DEST AND; THANKFULLY; NO LOSS OF SEPARATION OCCURRED. WE WERE ON AN OVERNIGHT TRIP; HOWEVER; I WAS WELL RESTED WITH ABOUT 7 HRS OF SLEEP THAT NIGHT AND NO REMAINING SLEEP DEFICIT. HOWEVER; BOTH OF US WERE DISTR AS WE TOOK THE RWY. THE KNOB TO ADJUST MY SUN VISOR HAD BROKEN OFF PREVIOUSLY; AND CONSEQUENTLY ANY ADJUSTMENTS HAD TO BE MADE WITH A DIME INTO A SLOT SCREW TO MOVE IT AROUND; AS I HAD TO DO FOR THAT TKOF. I ENDED UP DROPPING THE DIME ON THE FLOOR AND SEARCHING LIKE MAD TO GET IT BEFORE TKOF. THE CAPT HAD TURNED AROUND I BELIEVE TO ANSWER A LAST MIN QUESTION FROM A PAX. WHEN EACH OF US WAS DONE; WE COMMENCED THE TKOF ROLL AFTER OUR LINE-UP CHKS. I BELIEVE THAT SINCE EACH OF US WAS DISTR; WE BOTH THOUGHT THE OTHER WAS PAYING ATTN TO ATC; AND SINCE NO ONE QUESTIONED THE LACK OF CLRNC; WE EACH ASSUMED THE OTHER ONE 'HEARD' IT.