I'm going to use an example from an anime I watch, Ouran High School Host Club. There are a pair of twins named Kaoru and Hikaru. Together they're pretty mischievous and similar in how they act. However, once they're apart they start to develop independent personalities. Kaoru is actually kinder and more sympathetic when away from Hikaru, and Hikaru is actually a little meaner when apart from Kaoru. Each influences the other when close, but Hikaru -the cruel prankster- clearly has more sway when they're together, and thus he's more dominant. Having a linked soul like that means they can communicate with each other easier than regular people.

I hope that makes sense. ^^'