Well in the anime the Elite Four and Gym Leaders seem to hold power like celebrities in some cases. But in others, they are helpful members of the community, but only to the citizens of the town and people who have met them. Otherwise, like in the case of Brock and Misty, people had no idea that they were gym leaders.
But meanwhile, there are town officials, such as mayors and policemen and stuff. For instance, the mayor of Hearthome city ran the double battle tournament. And if the pokémon world (which is based off of Japan so far) was in Anarchy, then it would be much less organized. If there was no government then the festivals and contests that are held, basically every time a main character comes to a town, would not take place.
And how are the mayors elected? There must be a similar democratic system to the one currently in Japan.