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Somewhere in the middle would seem to be religions like Buddhism which, though large parts of their theology may have to be reworked in order to bring them in line with the new knowledge of intelligent life elsewhere in the universe, their general teachings would not have to change very much in order to be reconciled with this new knowledge.
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From what I understand of Buddhism in general and Tibetan Buddhism in specific, based on personal studies and Powers's text
Introduction to Tibetan Buddhism, the existence of off-world religions, alien gods, and extraterrestrial life in general wouldn't have much of an effect at all on "how Buddhism works." In Tibetan Buddhism (which came from India via Nepal), the "deities" (for lack of a better term) are mostly localized spirits that were "elevated" to godlike status as the belief system evolved. At its simplest state, the practice of Buddhism doesn't require belief in ANY deities of any kind, whether earthly or otherwise; it is a wholly internalized system, based in understanding one's self in relation to the world in which one exists. The Buddha didn't preach belief in gods, he spoke about how each person individually can transcend suffering and become enlightened. Later followers brought in the concepts involving "deities" from the local peoples, in order to make the dharma more accessible to the people of those areas. Debate continues even today on whether Buddhism is a religion or a philosophy. My personal take is in line with those who hold that a religion requires a creator deity, of which there is none in Buddhist thought.
I'll stop there, because it's all too easy to just keep going like the Energizer Bunny. In sum, my take is: Alien life's existence and proof thereof to humans on Earth has (will have?) little if any effect on Buddhism. The search for enlightenment is internal to ourselves; what happens "out there" happens, whether we're on the path to enlightenment or not.