The word “people” functions as an aggregate noun here. This site from Georgia State U says that ”Many aggregate nouns have plural forms, but many also are not plural in form, though they use plural verb forms. Example: The police are still looking for the Olympic Park bomber.”
Wikipedia has a page on people wouldn’tcha know it that says that usage includes: the plural of person (in addition to the rarer plural, ‘persons’) or a group of people (grammatically, a suppletive plural and collective noun; e.g. ‘some people are…’) ...
Having established that “people” is a plural, it seems reasonable that brains would also be plural, there generally being a one-to-one correspondence between people and brains. Unless this is a sci-fi scenario where people are tapping into a single, collective group brain as pointed out by the OP.
Since “brains” is plural, it takes the plural possessive pronoun “their.”
Had it been “brain” (singular), there is controversy about the usage of the gender-neutral “their” instead of the phrase “his or her”, as in “Each individual should use their brain.”
Ambiguity arises, I think, because “brains” can also be a collective singular for cranial contents, as in “He blew his brains out!” Or “She’s got lots of brains,” meaning: She is intelligent; not meaning: She is Igor’s assistant and is carrying several brains from the lab.