Client Chat Listener Mixin

InGameHudAccess.java

package com.packagename.access;

import net.minecraft.client.gui.ClientChatListener;
import net.minecraft.network.MessageType;

public interface InGameHudAccess {
    void registerChatListener(MessageType messageType, ClientChatListener listener);
}

InGameHudMixin.java

package com.packagename.mixin;

import com.packagename.access.InGameHudAccess;
import net.minecraft.client.gui.ClientChatListener;
import net.minecraft.client.gui.hud.InGameHud;
import net.minecraft.network.MessageType;
import org.spongepowered.asm.mixin.Final;
import org.spongepowered.asm.mixin.Mixin;
import org.spongepowered.asm.mixin.Shadow;

import java.util.List;
import java.util.Map;

@Mixin(InGameHud.class)
public class InGameHudMixin implements InGameHudAccess {

    @Shadow
    @Final
    private Map<MessageType, List<ClientChatListener>> listeners;

    public void registerChatListener(MessageType messageType, ClientChatListener listener) {
        this.listeners.get(messageType).add(listener);
    }
}

MyModEntrypoint.java

package com.packagename;

public final class MyModEntrypoint implements ModInitializer {

    @Override
    public void onInitialize() {
        Logger LOGGER = LogManager.getLogger("TestMod");
        // Wait for CLIENT_STARTED, otherwise `inGameHud` will be null.
        ClientLifecycleEvents.CLIENT_STARTED.register(client -> {
            ((InGameHudAccess)client.inGameHud).registerChatListener(
                MessageType.CHAT,
                (type, message, sender) -> {
                    LOGGER.info(String.format("%s:: %s", sender.toString(), message.toString()));
                });
        });
    }
}

your_mod_name.mixins.json

{
  "required": true,
  "package": "com.packagename.mixins",
  "compatibilityLevel": "JAVA_17",
  "mixins": [
    "InGameHudMixin"
  ],
  "injectors": {
    "defaultRequire": 1
  }
}

Good, thank you!