How to reward agent's action in self-play adversarial game reinforcement learning?

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I am beginner in this field and I am trying to implement an agent playing an adversarial game like chess. I create two agents that share the same neural network and experience buffer. In every step, the neural network will be updated by both agents (features order swapped).

Does my self-play approach make sense? And if it does, how will I reward the agent's behaviors?

More clearly, following this:

(0) state -> (1) agent0 action -> (2) reward -> (3) state -> (4) agent1 action -> (5) reward -> (6) state

Is the next state of agent0 after (1) would be (3) or (6)? And is the corresponding reward (2) or (5) or something else (for example (2) - (5))?

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