So the main point of this question came up when I was discussing with my friend things that affect someone’s overall leadership score. Obviously, no character should be allowed to get two totals for minions and cohorts, yet each persona could have different modifiers for things like fame, treatment, etc. Therefore, would it make more sense to have the base score, use that for both personas, and then apply situational modifiers separately, or to apply the feat to only one persona? In the case of the feat being applied to one persona, would it make sense to allow the character to take Leadership or similar feats more than once total, one for each persona?
These questions are especially pertinent to someone like a vigilante with a class feature that innately grants a dual life nature.
A cohort will almost certainly know both personas by the nature of their closeness. Followers maybe not, but in that case you have to go with the reality that followers (or even the oblivious cohort) would only follow the persona that leads them, because the other persona simply isn't that same person in their eyes.
Make the "Vigilante" pick a persona that gets the followers/cohort (if the cohort doesn't know), then that persona's reputation is the only one that matters.
Since you only get a single cohort, the modifiers are usually not going to make a difference. The only thing it might affect is the level of your cohort. Most character taking leadership have a decent CHA, so the score is max level of the cohort is 2 less than your character level. Unless the character has multiple penalties, chances are you will have a cohort of 2 less than your level. If the cohort is aware that the character is the same person, they won’t have a problem obeying both personas.
The number of followers and their level can vary a lot depending on the persona. Since the vigilantly came out much later than the feat there is nothing in the rules on how to handle this. This is going to be in the realm of ask your GM.
If the followers are also aware of the characters separate personalities, I would give the character the benefit of the doubt and allow him to use the better of the two modifiers. That is total up the modifiers for both personalities and use the better of the two.
If the followers are not aware of the separate personalities it is going to depend on how the character wants to have followers. If the character wants to only have followers in one personality, they can simply choose which personality and use the modifiers for that personality. If the character wants both personalities to attract followers, I would have them track their leadership scores separately and give half the numbers listed rounding down.
Again, this is the way I would handle it in a campaign I run.