Their stories are separate but it's all on one playthrough that you can switch between. You can go through the characters' stories and chapters in whatever order you want, but the levels do jump between each next chapter. So unless you wanna do a ton of grinding, you'll probably need to go through everyone's chapter 1s, then go onto chapter 2s, etc.
Your main character/who you picked first is locked into your party, and you'll need to have the character of whose story you're going through in your party, as well.
I don't have a copy of full skills/abilities list on hand, but someone else might.
