One-time on this device. The proxy checks it server-side before it relays anything; your own Duels.ink token is separate and entered per sync.
Your token is used for this sync only: it goes to the proxy, the proxy relays to Duels.ink, and it is never stored anywhere. Replays fetch at ~25/min (their hard limit is 30), so a first sync of 100 games takes a few minutes. Re-syncs only fetch games not already on this device.