" Response: After you engage an enemy with a higher engagement cost than your threat, ready Sam Gamgee. He gets +1 Willpower, +1 Attack and +1 Defense until the end of the round. "

Those +1 stats persist every round as long as the enemy is engaged with us? Or only the round it engages us?

His response only triggers when an enemy goes from being not engaged to engaged, so it'll only kick in for that first round.

After that, the enemy is already engaged so the response doesn't happen.

If however you used a card effect to un-engage the enemy (A Light In The Dark, for example), then when you engage the enemy a second time, the response would kick in.

Also note that he gets the +1 to the stats even if he is not readied by the effect.