Seabed Trawl

It happens for the player who claims it, whoever launched it. A claim happens in the battle step of its claimer's own turn (706.3a), so it waits for that turn's second watch, and happens as it starts (706.6a). Look at the top 2 cards of your hold. Draw one; put the rest on the bottom.

Crew dispatched here on their own turn make progress, whoever launched it; on the opponent's turn they can only fight (706.3a). The first player to reach 2 progress claims it and gains 1 spoils.