Requirements:
1)Defeat Misaki or Beat the Game
2)Have talk to Geng
3)Have the Fragments

Here are the Legendary Monsters:

Rainbeau=Must have Rainbow Fragment and then go to the shrine on Great Barrier Reef on Oceania
Typhorn=Must have Mystic Fragment and then go to the shrine on Taklamakan on Asia
Kristar=Must have Blizzard Fragment and then go to the shrine on North Pole on Eurasia
Snyph=Must have Dark Fragment and then go to the shrine on Giza on Africa
Wreckx=Must have Earth Fragment and then go to the shrine on Grand Canyon on North America
Slythe=Must have Wisdom Fragment and then go to the shrine on Machu Pichu on South America
Furion=Must have Flame Fragment and then go to the shrine on Mt.Etna on Europe

((((((((=I HOPE IT HELPS=)))))))))))