FLC OF A B727 TOOK OFF OVER AN AIR TAXI ACFT WHICH WAS ANNOUNCING OVER CTAF THAT HE WAS TAKING THE RWY AT A TXWY INTXN FOR A TKOF IN THE OPPOSITE DIRECTION TO WHICH THE RPTR IN THE B727 WAS TAKING OFF.

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DUE TO A MECHANICAL PROB; OUR FLT DEPARTED CID AFTER THE TWR CLOSED FOR THE NIGHT. WE COMPLIED WITH OUR COMPANY'S 'NON TWR' PROCS FOUND ON OUR COMMERCIAL CHART PAGE. THE FO CALLED ZAU PRIOR TO TAXI FOR OUR CLRNC USING THE VHF #2 RADIO. THE FO THEN MADE A 'TAXIING TO RWY 9' CALL ON THE CTAF USING THE VHF #1 RADIO. UPON REACHING THE END OF RWY 9; THE FO CALLED 'TAKING THE ACTIVE' ON CTAF. AS WE LINED UP; WE HEARD ACR Y ANNOUNCE HIS TAXIING ON THE CTAF. THE FO THEN CALLED 'BEGINNING TKOF ROLL' ON CTAF AND ZAU'S FREQ. I XFERRED ACFT CTL TO THE FO AND WE BEGAN ROLLING. BTWN 80-100 KTS; WE HEARD ACR Y ANNOUNCE HIS INTENTIONS FOR AN INTXN TKOF (TXWY A) ON RWY 27. THE FO AND I BOTH HAD THE ACR Y IN SIGHT; AND AS WE NEARED V1 (ROTATE SPD) WE SAW THAT HE WAS NOT GOING TO STOP PRIOR TO ENTERING THE RWY. THE FO EXPEDITED ROTATION AND WE WENT OVER THE ACR Y BY APPROX 300 FT AS HE ENTERED THE RWY. DURING THIS TIME; I TRIED CONTACTING ACR Y ON CTAF BUT HE DID NOT RESPOND. SOON AFTER CLRING HIM; WE HEARD HIM QUERY ZAU IF THEY JUST HAD SOMEONE TAKE OFF FROM CID. ZAU TOLD HIM OF OUR DEP; AND ACR Y WANTED TO KNOW WHY WE HADN'T MADE ANY CALLS ON THE CTAF. WE TOLD ZAU THAT WE HAD IN FACT MADE CALLS THAT WENT UNANSWERED BY ACR Y. I ALSO TOLD ZAU WE HAD HEARD ACR Y BUT HE EVIDENTLY HADN'T HEARD US. ACR Y SAID SOMETHING ABOUT THE VOLUME ON HIS RADIOS; AND THAT 'HE GUESSES HE'S JUST USED TO HAVING CID TO HIMSELF AT THAT TIME OF NIGHT' ZAU RESPONDED THAT IT SOUNDED LIKE EVERYBODY DID WHAT THEY WERE SUPPOSED TO HAVE DONE. ONE THING; HOWEVER; WAS NOT DONE. I CAN UNDERSTAND MISSING SOME RADIO CALLS; BUT I CANNOT UNDERSTAND HOW SOMEONE TAXIES ONTO A RWY WITHOUT VISUALLY CLRING IT -- ESPECIALLY WHEN THE TWR IS CLOSED.