One fear people have of alien contact is how the proof of alien existence will affect our theologies; this includes how, beyond "just" their existence, alien theologies will affect our theologies.
One thought that popped in my head the other day was what if an alien race had in their theology that they, as a people, would corrupt and lose their true religion - their church would be without leadership, without a head, and that one day a new church leader would come, one who at first did not know they were appointed to be their leader, one who was from another race, one extrasolar and thus alien to them, who would come from the stars, learn their religion and lead their church. How would that affect us?
If their mythology predicted their church would become corrupt and they would lose all knowledge of the true religion and that someone from the heavens would come to show them the right way - that would be a call to action for many, if not most, of the churches on Earth to send missionaries and possibly fight over who would lead the aliens to the light (as Earthlings see it).
But in the first scenario - a human would need to go to the alien planet, learn their religion, and lead them. Would that cause the alien religion to be adopted by some people on Earth? Would they adopt it because it essentially elevates humans as saviors? Would they adopt it because it presents the promise of power? Would they adopt is because they have become cynical of Earth religions? Would they adopt it because they would feel accepted, part of something greater that they were not able to get, for whatever reasons (good or bad or imagined), in churches here on Earth?