1 MLG TOOK OFF FROM RWY 19L WITHOUT CLRNC WHILE ANOTHER MLG TOOK OFF FROM RWY 25R WITH PROPER CLRNC. A VERY CLOSE CONFLICT RESULTED.

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WE WERE GIVEN INSTRUCTIONS TO TAXI INTO POS AND HOLD BY THE TWR CTLR. AFTER APPROX 15 TO 20 SECONDS; WE WERE GIVEN 'MAINTAIN RWY HDG; CLB TO 7000 FT' AND WHAT WE UNDERSTOOD WAS A CLRNC FOR TKOF. WE THEN PROCEEDED WITH THE TKOF AND AFTER BECOMING AIRBORNE NOTICED A B-737 CLBING OUT ON RWY 25R. WE MADE A R TURN TO THE W TO AVOID THE OTHER ACFT'S DEP TRACK; AT WHICH TIME WE WERE TOLD TO TURN R BY THE TWR CTLR. SHE ALSO ADVISED US THAT WE WERE NOT GIVEN A TKOF CLRNC. THE FLT PROCEEDED TO SAN DIEGO NORMALLY. SUPPLEMENTAL INFO FROM ACN 267879: WE WERE CLBING OUT ON DEP FROM (RWY 25R) LAS TO SFO WHEN OUR TCASII ALERT SOUNDED AT APPROX 600 AGL. I WAS FLYING IN THE R SEAT AND SCANNED OUT THE R SIDE WINDOW WHERE THE INTRUDER (TCASII) SHOULD HAVE BEEN. I SAW A 737 APPROX 300 YARDS AWAY CLBING OUT ON A RWY 19L TKOF COMING DIRECTLY AT US. TWR SAID SOMETHING TO THE EFFECT THAT 'ACR X; YOU ARE NOT CLRED FOR TKOF.' I IMMEDIATELY ROLLED L INTO A 45-50 DEG BANK AWAY FROM ACR X AND I BELIEVE HE BEGAN A R TURN EVASIVE ACTION AT THE SAME TIME. WE WERE AT ABOUT 800-1000 FT AGL WHEN I BROKE L. I BELIEVE THAT ATC COULD HAVE BEEN MUCH MORE AGGRESSIVE IN GIVING COLLISION AVOIDANCE GUIDANCE SOONER TO US AND ACR X. ALSO; AFTER WE HAD BEEN CLRED FOR TKOF AND DEPARTED I HEARD A COUPLE OTHER XMISSIONS BTWN THE ACR X 737 AND ATC THAT MAY HAVE CONFUSED THE ACR X CREW. I WAS BUSY WITH THE TKOF PROCS AT THE TIME AND DON'T REMEMBER EXACTLY WHAT WAS SAID. SUPPLEMENTAL INFO FROM ACN 268219: THE OTHER B737 HAD APPARENTLY TAKEN OUR CLRNC FOR TKOF AS HIS AND TOOK OFF ON RWY 19.