To add to kersing’s questions:
Is this relevant for what you want, or do you want all messages (for any node) that the gateway “sends”?
If you only want to know about this specific device, which might actually also be handled by other gateway(s), then the MQTT Data API (including its #/events/# topic) might suffice to see what’s going on?
Is that “sends” as in “LoRaWAN downlinks that it has been commanded by TTN to transmit wirelessly”, or as in “LoRaWAN uplinks that it forwards to TTN over the internet”? Or is it even about the bare packets of the LoRa radio protocol that are being received and transmitted, rather than the bytes of the LoRaWAN messages?
Without setting up some private network, to see what the TTN Gateway is doing for devices that are not yours, I think you’ll need to compile your own firmware to log all received and transmitted LoRaWAN packets. If you’re after the LoRa radio packets, then I think it cannot be done on the gateway, as the gateway delegates all that to the embedded LoRa card which you cannot easily reprogram yourself.