A PA28-161 INSTRUCTOR PLT ALLOWS AN UNAUTH TKOF FROM MLB; FL. AN NMAC WITH A DEP CHALLENGER ENSUES DURING THE INITIAL CLB.

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MYSELF IN THE R FRONT SEAT; A PVT STUDENT IN THE L FRONT SEAT AND A PVT STUDENT IN THE REAR SEAT; HAD FINISHED OUR RUNUP AND TAXIED TO THE HOLD SHORT LINE OF RWY 4 FOR TKOF. OUR FIRST CALL TO TWR WAS INTERRUPTED TWICE BY OTHER ACFT ON THE FREQ. IT WAS QUITE A BUSY EXCHANGE FOR THE CTLR; HE DID NOT SEEM ABLE TO TALK FAST ENOUGH BEFORE SOMEONE ELSE WOULD CALL IN. WHEN WE DID GET A HOLD OF HIM; WE WERE TOLD TO HOLD SHORT; WHICH WE DID. WHILE WE WAITED; I WAS REVIEWING WITH THE STUDENTS THE PROC FOR THE SOFT FIELD TKOF WE WOULD BE DOING. I HEARD THE CTLR USED; ONLY THAT IT SOUNDED LIKE THE SAME CLRNC FOR TKOF THAT WE HAD RECEIVED SO MANY TIMES BEFORE. THE STUDENT IN THE BACK SEAT WAS TALKING OVER THE INTERCOM AT THE TIME; SO I ASKED THE STUDENT CTLING THE ACFT IF WE WERE CLRED FOR TKOF AND HE SAID YES. WE PROCEEDED WITH OUR TKOF ROLL AND JUST BEFORE REACHING TKOF SPD; THE STUDENT LOST DIRECTIONAL CTL. THE ACFT STARTED TO SKIP OFF THE L SIDE OF THE RWY WITH AN EXTREME NOSE HIGH ATTITUDE; WITHOUT SUFFICIENT AIRSPD TO CLB. I HAD TO WRESTLE WITH THE STUDENT WHILE ALSO YELLING TO REGAIN ENOUGH CTL TO PREVENT A GND LOOP AND/OR HITTING THE RWY LIGHTS. I MANAGED TO LOWER THE NOSE ENOUGH TO GET THE AIRSPD NEEDED TO CLB. DURING THE NEXT FEW MOMENTS OF THE CLBOUT; I WOULD GUESS FROM ABOUT 50 FT TO APPROX 300+ FT; I DID NOT HEAR ANYTHING FROM THE TWR. I THINK THAT THERE HAD BEEN SOME RADIO TFC DURING OUR 'EMER;' BUT DUE TO THE URGENCY OF MY CURRENT SIT AND WORRIED ABOUT OUR ACFT AND OURSELVES; I DON'T REMEMBER HEARING ANYTHING DISCERNABLE BUT THE STALL WARNING HORN AND A JUMBLE OF DIFFERENT VOICES. AT APPROX 300-400 FT; THE CTLR CALLED US AND ASKED IF WE REALIZED THAT WE JUST TOOK OFF WITHOUT A CLRNC AND THAT WE MADE A CHALLENGER JET EXPEDITE HIS CLB TO MISS US. HE SAID TO REMAIN IN THE PATTERN AND RETURN TO LAND AND THEN CALL THE TWR. THAT IS WHAT WE DID NEXT. I DID NOT EVEN SEE THE JET UNTIL HE HAD TOLD US ABOUT IT. AT THAT POINT IT WAS OVER 1 MI AWAY. IT HAD TAKEN OFF RWY 9R AND WE WERE ON RWY 4. THEY BOTH MEET AT THEIR DEP ENDS. LOOKING BACK NOW AT THE SIT; I CAN SEE HOW THE BUSY TRAINING ENVIRONMENT AND ALL THE RADIO CALLS WERE DISTRACTING FOR BOTH ME AND THE CTLR. THE IMMEDIATE 'EMER' WE HAD ON THE RWY PROBABLY POLARIZED ME INTO DEALING WITH THAT PROB AND SHUTTING ALL OTHER THINGS OUT. (DURING OUR PHONE CALL THE CTLR SAID THAT HE TRIED TO 'CANCEL OUR CLRNC' WHEN WE 'BOBBLED OFF THE RWY;' BUT DID NOT GET A RESPONSE.) UPON DEBRIEF WITH THE STUDENTS; THE PLT SWEARS HE HEARD THAT HE WAS CLRED FOR TKOF. THE BACK SEAT STUDENT SAID HE SAW THE JET IN THE AIR BUT DID NOT SAY ANYTHING UNTIL IT HAD ALREADY PASSED OVERHEAD. (THE CTLR DURING OUR PHONE CONVERSATION ALSO MENTIONED HEARING; '...CLRED OR CLR (SOMETHING) ...;' IN THE BACKGND ON THE TAPE; BUT SAID THAT HE DIDN'T SAY IT.) A FEW OF THE OTHER FLT INSTRUCTORS ON THE FREQ AT THE TIME; WHO CAME UP TO ME AFTERWARDS TO ASK WHAT HAPPENED; SAID THEY WERE CONFUSED WHEN THEY HEARD THE CTLR SAY THAT OUR 'CLRNC WAS CANCELED;' SINCE HE CLAIMS NOT TO HAVE GIVEN ONE. I GUESS UNLESS I HEAR THE TAPE OF THE EVENT I WON'T KNOW FOR SURE. I DO KNOW THAT A LITTLE BIT MORE ATTN TO 'THE BIG PICTURE' OF WHAT WAS HAPPENING AROUND US COULD HAVE PREVENTED THE ENTIRE SIT.