With loyal the second protagonist will always agree with your decisions. With random, the second protagonist may or may not agree, in a random fashion and not according to any personality or pattern. With no AI you will be required to choose both dialogue options (effectively role-playing both characters).
If you want some interesting disagreements but don't want to roleplay both characters yourself, you may like to set both protagonists to random (in case you are controlling the other charcater at some point).
Further AI personality settings will come soon in future patches.
"Love one another and you will be happy. It's as simple and as difficult as that" - Leunig
