Normally, the requirement to unlock someone's Gaiden is to clear his/her Musou Mode. However for some characters, you need to clear another character's Musou Mode (Sometimes including Gaiden) as well to unlock them.

Effect - Task

Ishida Mitsunari's Gaiden - Beat Naoe Kanetsugu's Musou Mode as well
Oichi's Gaiden - Beat Nouhime's Musou Mode including Gaiden as well
Saika Magoichi's Gaiden - Beat Date Masamune's Musou Mode as well
Shimazu Yoshihiro's Gaiden - Beat Sanada Yukimura's Musou Mode including Gaiden as well
Uesugi Kenshin's Gaiden - Beat Takeda Shingen's Musou Mode including Gaiden as well