MLG GIVEN TRAFFIC INFORMATION TKOF CLRNC. WHEN LIFTING OFF TWR GAVE SECOND TKOF CLRNC TO MLG CAUSING CONCERN.

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WE MLG X WERE CLRED INTO POS AND HOLD ON RWY 18R AT MEM. AFTER TAKING THE RWY AND HOLDING IN POS FOR A SHORT PERIOD (30 SEC - 1 MIN) WE WERE ADVISED BY TWR THAT AN 'SMT Y PASSING LEFT TO RIGHT OVER DEP END OF RWY-CLRED FOR TKOF.' WE BOTH OBSERVED THE TFC AND ACKNOWLEDGED TFC AND TKOF CLRNC TO EACH OTHER AND TWR AND MADE THE TKOF. JUST AFTER GEAR RETRACTION THE TWR SAID 'MLG X CLRED FOR TKOF.' F/O ADVISED TWR WE HAD BEEN CLRED FOR TKOF AND WERE AIRBORNE. TWR THEN SIMPLY TOLD US TO CONTACT DEP CTL. DUE TO THE FACT THAT WE WERE BOTH ABSOLUTELY POSITIVE OF HAVING GOTTEN TKOF CLRNC AND VOLUME OF TFC ON TWR FREQ WE WENT TO DEP CTL AND NO FURTHER MENTION OF THE MATTER WAS MADE BY US ON ATC. WE HAD SMT Y TFC IN SIGHT AT ALL TIMES THROUGH DEP AND INITIAL CLB AND IT NEVER WAS A FACTOR. IT IS ALWAYS A BIT UNCOMFORTABLE IN THESE SITUATIONS WHERE THE CTLING FACILITY IS NOT CERTAIN OF 'WHO'S ON FIRST' AND THE NORMAL SCAPEGOAT OF ANY MISUNDERSTANDING IS THE COCKPIT CREW. THIS CAN DETRACT FROM THE PRIMARY DUTIES OF FLYING THE MACHINE.