FLC OF LGT ACFT INADVERTENTLY DEV FROM ASSIGNED HDG DURING DEP CLB.

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WE WERE CLRED FOR TKOF ON RWY 29L TO FLY RWY HDG AND MAINTAIN 5000 FT. AFTER TKOF I NOTICED THAT THE HDG BUG HAD NOT BEEN SET TO RWY HDG. THIS WAS NOT A PROBLEM IN ITSELF BUT WAS A SLIGHT DISTR DURING A BUSY PHASE OF FLT. AT ABOUT 1500 FT MSL (700 AGL) I SAW THE CAPT REACH UP TO THE HDG KNOB TO SET THE HDG BUG. AT THE SAME TIME TWR HANDED US OFF TO DEP. THE CAPT CALLED DEP; THEN I HEARD THE CAPT READ BACK 'TURN R TO 360' AND HE SET THE HDG BUG TO THIS HDG. I BEGAN A SLOW TURN TO 360. OUT OF ABOUT 2500 MSL THE TCASII SOUNDED 'MONITOR VERT SPD' AND INDICATED A RED ARC ON THE VSI. THERE WAS A RED RA SQUARE ON THE VSI SLIGHTLY BEHIND AND TO THE R OF OUR AIRPLANE SYMBOL. IT INDICATED THAT THE TFC WAS ABOUT 1000 FT BELOW US; CLBING. THE SYMBOL CHANGED TO A YELLOW TA SYMBOL BUT RELATIVE POS AND ALT SEPARATION STAYED APPROX THE SAME. WE THEN HEARD ANOTHER ACFT ASKING THE DEP CTLR ABOUT AN ACFT THAT HAD FLOWN OVERHEAD. EVENTUALLY WE WERE GIVEN A L TURN AND THE TA CIRCLE DISAPPEARED. THE CAPT THEN ASKED THE DEP CTLR IF THERE WAS A PROBLEM AND HE STATED THAT WE HAD TURNED N RATHER THAN CONTINUING NW. THE OTHER ACFT APPARENTLY TOOK OFF FROM 29R AS WE DEPARTED 29L. I BELIEVE ONE FREQ WAS HANDLING BOTH RWYS. I BELIEVE THE 360 HDG WAS FOR THE OTHER ACFT ALTHOUGH I DON'T KNOW IF THE CTLR USED OUR CALL SIGN BY MISTAKE OR IF THE CAPT MISHEARD THE CALL SIGN. IN ANY EVENT THE CAPT READ BACK THE CLRNC AND WAS NOT CORRECTED BY THE TWR CTLR. DETERMINING CAUSE AND POSSIBLE SOLUTIONS HINGE ON WHETHER THE CTLR USED THE WRONG CALL SIGN OR WHETHER THE CAPT RESPONDED TO SOMEONE ELSE'S CALL SIGN. HOWEVER; I BELIEVE THE MISSET HDG BUG; BY CREATING A DISTR DURING A BUSY TIME; WAS A FACTOR.