There are many changes to Great People mechanics, compared to Civilization V. First, each type of Great Person (including Great Prophets!) is earned via accumulating a distinct Great Person Point resource. Each civilization accumulates the relevant resource separately, but unlike in Civilization V, the accumulation happens empire-wide rather than citywide. Every distinct source of the same type of Great Person points will contribute to the common pool. Whenever a civilization attracts the Person (whether by acquiring enough points, or in other ways - see below), the pool is depleted and starts accumulating again.