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From what I've read in mechanism design theory, it seems to me that the mechanism designer always plays an essential role in a mechanism. Information is sent to the designer who designs incentives that aim to enforce the truthful revelation of information from the strategic agents of the mechanism.

But, within the mechanism design theory, is it possible to remove the requirement of agents communicating their information to a central authority (aka the designer)?

For example, design a framework where the agents communicate with each other based on certain rules and figure out by themselves the designer's desirable outcome without any communication between the agents and the designer.

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I am not sure if this is the answer you were aiming at, but... what about the revelation principle?

There is no requirement to communicate with the designer. However, researchers are ofter first interested in what the optimal allocation rule is. In that search, it is wlog to consider direct mechanisms by the revelation principle: Suppose an indirect mechanism without communication to the designer implements the optimal allocation, then we can obtain the same allocation if every agent simply reports their type to the mechanism designer who then computes the equilibrium allocation of this indirect mechanism and implements it. See my reply with more details here.

If you want to know if there is a general approach of how to find equivalent indirect mechanisms for the optimal direct revelation mechanism, I am not aware of an answer.

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