CTLR #1 MAY HAVE BEEN IN TRAINING ON LCL AT TEB CLRED A C750 INTO POS. A GLF ON FINAL WAS ISSUED A MISSED APCH BY CTLR #2. CTLR #1 THEN ISSUES THE C750 TKOF CLRNC. AT LIFTOFF CTLR #2 TELLS THE CREW THEY WERE NOT CLRED FOR TKOF.

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CTLR #1 HAD CLRED US TO 'POS AND HOLD' ON RWY 6. WHILE WAITING IN POS WE HEARD A GULFSTREAM RPT ON FINAL AND RECEIVE TWR ACKNOWLEDGEMENT. AFTER ABOUT 1 MIN WE HEARD THE GULFSTREAM ASK TWR IF HE SHOULD GO AROUND. THERE WAS NO RESPONSE FROM THE TWR. A FEW SECONDS LATER WE HEARD THE GULFSTREAM ASK AGAIN IF HE SHOULD GO AROUND. I REACHED OVER AND DISABLED SQUELCH ON COM #2 (OUR ACTIVE VHF COM) AND HEARD A DIFFERENT CTLR (#2) GIVING ALTERNATE MISSED APCH INSTRUCTIONS TO THE GULFSTREAM. TWR AUDIO SOUNDED WEAK SO I DIALED IN TWR FREQ ON COM #1. THE SIC AND I BOTH SWITCHED AUDIO CTL TO COM #1. TWR AUDIO WAS NOW CLR. WE HEARD CTLR #1 ISSUE TKOF CLRNC TO A HELI AND A FEW SECONDS LATER CANCEL THE CLRNC. AN ARPT VEHICLE REQUESTED SOMETHING; WHICH NEITHER I OR THE SIC COPIED CLRLY; WHICH WAS APPROVED BY ONE OF THE 2 CTLRS (I DON'T RECALL WHICH). 10-20 SECONDS LATER WE RECEIVED TKOF CLRNC FROM CTLR #1; WHICH THE SIC ACKNOWLEDGED AND WE BEGAN OUR TKOF. AT APPROX 110 KTS AND HALFWAY DOWN THE RWY I SAW A PICKUP TRUCK SPDING ACROSS THE DEP END OF THE RWY. AT THE SAME INSTANT CTLR #2 SAID 'ACFT X WHAT ARE YOU DOING? YOU WERE NOT CLRED FOR TKOF.' I TOLD THE SIC WE WERE CONTINUING THE TKOF. WE WERE WELL IN EXCESS OF THE 80 KTS WE HAD BRIEFED AS OUR 'ABORT FOR ANYTHING' SPD. THE TRUCK HAD CLRED RWY AND ABORTING THE TKOF AT THAT POINT POSED A GREATER THREAT THAN CONTINUING. THE GULFSTREAM HAD OVERFLOWN US PRIOR TO OUR DEP AND WAS WELL CLR OF OUR DEP PATH. THERE WAS NO OTHER RPTED TFC. THE SIC TOLD TWR WE WERE CONTINUING TKOF. TWR HANDED US OFF TO DEP WHO ADVISED US TO CALL TEB TWR WHEN WE LANDED. OUR CHIEF PLT CONTACTED TEB WHO ADMITTED THIS WAS THEIR MISTAKE (A TRAINEE CTLR).