Each Legendary weapon must be acquired by using the Celestial Mirror.

Tidus: Caladbolg
Race and defeat the Chocobo trainer in the Calm Lands. After winning the race, walk to the upper northwest section of the Calm Lands. The guard blocking a narrow pathway down into the gorge will have left his post, allowing you to pass. Tidus will then be able to use the Calastial Mirror to acquire Caladbolg.

Yuna: Nirvana
Capture all nine types of fiends in the Calm Land region. After collecting them all, return to the monster arena owner. He will produce a treasure chest, containing the weapon.

Auron: Masamune
Acquire the Rusty Sword, which is found on the eastern cliff of the Cavern of the Stolen Fayth in the Calm Lands. Take the Rusty Sword to the statue of Lord Mi'ihen on the Mushroom Rock Road. Place the Rusty Sword before the statue to reveal a glyph. Touch it to reveal the Masamune's secret location.

Wakka: World Champion
Talk to the bartender in Luca Cafe. If she refuses to turn it over, it is because you have been neglecting to play Blitzball.

Kimahri: Spirit Lance
Active three of the Qactuar Stones on the Thunder Plains, then find the Qactuar ghost flying just above the ground at Thunder Plains-South. Follow it to a small ruined lighting tower on the right side. Press Square to send a bolt of lightning to the tower to reveal a treasure holding the weapon.

Lulu: Onion Knight
Return to Baaj Temple and dive underwater. Swim towards the doors to the north and fight Geosgaeno. After defeating it, search the southern part of the circular underwater area and look for a treasure chest containing the weapon.

Rikku: Godhand
Input the airship password, "GODHAND" to open a secret location at Mushroom Rock. Move north to the end of this ravine, and use the Celestial Mirror on the symbol embedded in the rock face.