NMAC BETWEEN ATX TKOF AND SECOND ATX ON LNDG APCH.

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CHARTER PAX FLT FROM LANAI (LNY); HI TO HILO (ITO); HI. FLT WAS PLANNED IFR V-2 7000' LNY TO ITO. PRIOR TO TAXI WE CALLED ATX XYZ TAXIING FOR DEP RWY 21 LNY. THEN UPON ENTERING RWY WE CALLED ATX XYZ BACK-TAXIING RWY 03 FOR DEP ON RWY 21 LNY. ON TKOF ROLL THE XMISSION WAS MADE DEPARTING RWY 21 LNY. ALL THESE XMISSIONS WERE MADE ON CTAF 122.9. AT ROTATE SPD OF 105 MPH THE COPLT NOTICED AND POINTED OUT TO ME AN ACFT APPEARING TO TURN ON TO FINAL APCH AT LNY FOR RWY 03. I COULD SEE NO LNDG LIGHTS; ONLY THE RED AND GREEN NAV LIGHTS. I ADVISED THE COPLT TO XMIT TO THE ACFT THAT WE WOULD MAKE AN IMMEDIATE LEFT TURN AFTER TKOF AND HEAD SE. NOTHING WAS HEARD FROM THE ACFT ON FINAL. I SWITCHED ON THE LNDG LIGHTS AS SOON AS I SAW THE OTHER ACFT BUT I DID NOT SEE HIS LNDG LIGHTS AT ANY TIME. I WAS ABLE TO CONTACT THE ACFT AFTER IT LANDED AND WAS ADVISED BY THE COPLT THAT SHE DID NOT HEAR US CALL FOR TAXIING OR TKOF BUT THAT THEY HAD CALLED ON BASE AND FINAL FOR LNDG AT LNY. IT IS UNKNOWN HOW CLOSE THE ACFT CAME WITHIN EACH OTHER; BUT IT IS KNOWN THAT THE ACFT ON FINAL DID NOT ATTEMPT A GO AROUND NOT DID ITS PLT FLY OVER THE RWY TO DETERMINE THE DIRECTION OF THE WIND. HAD AN OVERFLY BEEN ATTEMPTED; WE WOULD HAVE BEEN ALERTED THAT ANOTHER ACFT WAS IN THE AREA AND MORE SPECIFICALLY PLANNING TO LAND. IT WILL BE CONSIDERED PROC FOR ALL COMPANY PLTS TO PLAN NIGHT TKOFS WITH LNDG LIGHTS ON UNTIL GEAR IS RETRACTED. THE EFFECTS OF A SINGLE ENG FAILURE AT MINIMUM CONTROL SPD WITH LNDG LIGHTS EXTENDED MUST BE A FACTOR IN THIS PROC.