ACR LTT AND ACR MLG HAVE CLOSE CALL WHILE BOTH ARE ON BASE LEG TO THE SAME RWY; OPPOSITE DIRECTION.

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LTT X ON STRAIGHT IN APCH TO RWY 27 WHEN AN LGT; INSTEAD OF TURNING OFF AT THE NORMAL HIGH SPD DECIDED TO USE THE WHOLE RWY FOR HIS LNDG ROLLOUT. SEEING THE LGT WAS NOT GOING TO BE CLR OF RWY 27 BEFORE WE CROSSED THE LNDG THRESHOLD; WE WERE TOLD TO GO AROUND AND TO ENTER A RIGHT DOWNWIND FOR RWY 1. AT THIS TIME THE GO AROUND (BALK LNDG) CHKLIST WAS BEING COMPLETED BY MYSELF. AS WE ENTERED DOWNWIND FOR RWY 1 I WAS ASKED BY THE CAPT IF WE WERE CLRED TO LAND. I SAID; 'LET ME CHK;' BECAUSE I WASN'T SURE. I THEN ASKED THE TWR IF WE WERE CLRED TO LAND RWY 1. TWR REPLIED; 'CLRED TO LAND; START YOUR BASE; NOW YOU'LL BE #1 INSIDE MLG Y.' EXACT POS WASN'T GIVEN. WE TURNED OUR BASE AND ABOUT A 245 DEG HDG. 15 DEGS BEFORE WE WERE PERPENDICULAR TO OUR LNDG RWY MLG Y TOLD THE TWR THAT THEY HAD AN ACFT COMING DIRECTLY AT THEM. WE WERE CURRENTLY XING THE CENTER LINE FOR RWY 1 DUE TO THE FACT THAT THE DOWNWIND LEG WAS SO TIGHT BECAUSE OF THE GO AROUND ON RWY 27. (MLG Y WAS PRESENTLY ON ABOUT A 075-080 DEG HDG). THE TWR TOLD US TO KEEP THE RIGHT TURN COMING AND THAT WE'D STILL BE INSIDE MLG Y; NOW IN SIGHT ABOUT 12:30-1 O'CLOCK; 1-2 MI. THE TWR THEN TOLD MLG Y TO MAKE A RIGHT TURN AND THAT HE'D BE #2 TO FOLLOW US. SEEING THAT WE BOTH HAD EACH OTHER VISUALLY AT THIS TIME; WE PROCEEDED TO PASS ON OUR RESPECTIVE HDGS; ESTIMATING ABOUT A 500' MISS. MLG Y WAS THEN CLRED TO LAND AND WE WERE GIVEN A LEFT TURN TO INTERCEPT FINAL FOR RWY 1 AND THEN CLRED TO LAND. THE WHOLE TIME IT APPEARED MLG Y DIDN'T KNOW WHO WE WERE; WHERE WE CAME FROM. FROM HIS REACTION HE WAS AS SURPRISED AS WE WERE. ONE POINT THAT MIGHT BE NOTED IS THAT WE WERE BOTH MODE C EQUIPPED AND FUNCTIONAL. THE AIR TFC LOAD AT THE TIME WAS LIGHT TO MODERATE; WX WAS VFR; CLEAR 20 MI VIS.