The only thing I can think of that this could refer to is split brain patients who have undergone a specific medial procedure once used to treat epilepsy. It's possible, although not well understood, that the two halves of their brain operate sort of independently, and only one has access to speech.
It's a very interesting topic, well worth diving into! But it's also very muddy, there is contrasting evidence for and against the "two mind" theory.
Anyway, I'm fairly certain this "theory" does not apply to healthy brains which have not undergone this procedure.