A B757 FLT ENCOUNTERS ANOTHER B757 AT THE INTXN OF 2 AIRWAYS AT FL350 OVER THE ESV VOR ON FREQ WITH SEGU; FO.

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LOCATION: ESV VOR. OVER ESP AT FL350 AT XY10Z; TCASII ISSUED 'TFC; TFC.' A YELLOW TARGET WAS DISPLAYED AT THE 3:30 O'CLOCK POS; 1-2 MI. ALMOST IMMEDIATELY A DSND RA INITIATED. WE FOLLOWED THE DSND RA; APPROX 1500 FPM. WE ACQUIRED A VISUAL ON AN ACR B757 AT THE TCASII LOCATION. I INITIATED A RADIO CALL THAT WE WERE RESPONDING TO A TCASII RA AND DSNDING FROM FL350. ACR ACKNOWLEDGED THE RADIO CALL AND STATED THAT THEY HAD A VISUAL ON US. GUAYAQUIL CTL THEN DIRECTED US TO DSND TO FL310. WE ESTABLISHED CONTACT WITH B757 ON FREQ 130.55; SHARING FLT INFO. B757 HAD DEPARTED LIMA ON UG437 AT FL350. WE WERE FLYING UW7 FROM MNV TO ESV AT FL350. FL350 HAD BEEN ISSUED TO US BY CLRNC DELIVERY ON THE GND AT GYE. WE PASSED MNV AT XX53 AT FL350; ESTIMATING ESV AT XY10. THIS WAS RPTED TO GYE. AT NO TIME WAS ANY TFC INFO PASSED TO US. THE WX AT THE TIME WAS CIRRUS CLOUDS WITH MARGINAL VISIBILITY. WE ESTIMATE OUR CLOSEST APCH TO THE B757 WAS 1-2 MI. SUPPLEMENTAL INFO FROM ACN 466137: APCHING ESMERALDAS VOR ECUADORIAN COASTLINE WE HAD A TA; THEN A DSND RA ON UW7 APCHING ESMERALDAS VOR WHERE WE INTERSECT UG437. FLT LIM-ATL WAS ALREADY ON UG437. WE WERE BOTH AT OUR CLRED ASSIGNED ALTS; FL350; ESSENTIALLY ARRIVING AT ESV AT THE SAME TIME. WE COULD SEE TRAFFIC OFF OUR R-HAND SIDE. TCASII WORKED FINE. WE DSNDED TO FL310 AT THE DIRECTION OF GYE CTLR AFTER WE TOLD HIM OF OUR DSCNT ALREADY IN PROGRESS. I AM SURE WE WERE CLRED TO FL350; AND CAPT RPTED SO AT OUR POS.