Gyms

Gyms are run by players, not by the server. A gym has a leader, and you can only challenge it while that leader has it open.

Challenging

Open the gym list and left click a gym to challenge it. If it is closed you will be told so, and there is nothing to do but come back later.

The leader can accept or reject your challenge. When the battle ends, the result is announced to the server either way, so a win is public and so is a loss.

Rewards

Beating a gym pays out whatever that gym has attached to it. Rewards are set per gym, so two gyms are not worth the same.