MD80 FLT CREW EXPERIENCES MULTIPLE ENCOUNTERS WITH ANOTHER ACFT WHILE ATTEMPTING DEP FROM UNCONTROLLED MTJ.

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WE WERE DEPARTING MTJ ON A CLR DAY WITH GREAT VISIBILITY. THERE WERE NUMEROUS LIGHT ACFT AND JET ACFT IN THE TFC PATTERN. MY FO WAS MONITORING BOTH CTAF AND ZDV. I WAS ONLY MONITORING CTAF. WE MADE ALL REQUIRED RADIO CALLS. I HAD NOTICED 2 ACFT FLYING APCHS TO RWY 31 AND HEARD 2 ACFT CALL FOR RWY 17. THE 2 FOR RWY 17 WERE A TWIN ENG ACFT CALLED 'EXECUTING MISSED APCH FOR ILS RWY 17;' (NEVER HEARD HIM CALL AGAIN AFTER THE INCIDENT SO I DID NOT GET HIS NUMBERS) AND A B737. WE WERE INSTRUCTED BY ZDV TO HOLD UNTIL WE GOT A VISUAL WITH THE B737. WE HAD PREVIOUSLY ADVISED CTAF (MONTROSE TFC) THAT WE WERE HOLDING SHORT OF RWY 35. THE B737 RPTED OVERHEAD THE ARPT AND HDG W FOR THE VISUAL TO RWY 17; WE TOLD DENVER WE HAD HIM IN SIGHT; DENVER THEN CLRED US TO DEPART. IT HAD BEEN OVER 3 MINS SINCE THE TWIN HAD ANNOUNCED HIS MISSED APCH. WE CLRED THE FINAL; CALLED INTO POS RWY 35 AND CHKED THE B737 AGAIN WHEN IN POS. AS WE STARTED OUR TKOF ROLL (JUST AT BRAKE RELEASE) MY VERY OBSERVANT FO SAID TO STOP AND LOOK AT THE TWIN ON SHORT FINAL. THE ACFT WAS WHITE IN COLOR AGAINST A WHITE SNOW BACKGROUND. THE TWIN EXECUTED A L-HAND (EBOUND) CLBING TURN AT THE APCH END OF RWY 17. AFTER THE ACFT WAS CLR WE FOUND THE B737 AGAIN AND MADE ANOTHER CALL DEPARTING RWY 35. THE TWIN THEN PROCEEDED TO TURN OVER THE TOP OF US SO WE HAD TO CONTINUE TO THE NW TO OUT CLB HIM AND DEPART TO THE E. THE TWIN ACFT NEVER MADE ANY RADIO CALLS TO MONTROSE TFC AFTER THE INITIAL MISSED APCH CALL.