It determines how the second protagonist behaves in multiplayer conversations. No AI means you choose the dialogue lines yourself. Loyal means the second protag will just automatically agree with the PC you're controlling. Random is, well, random.
More variety in AI personalities would be welcome: selfish, ambitious, altruistic, etc. Perhaps it's moddable.
And yes, a real personality for the second member would be great. I hesitate to start the game right now or wait for new personalities. It's an important feature to me.
I'm hoping they put in some real AI choices soon. I've been playing both sides myself, which works fine, but sometimes it's nice to feel like the other person has a personality of their own.