Wardens of the West's ability is a Reaction to winning any intrigue challenge, so it can be triggered on both offense and defense. The reaction ability is in addition to claim, and occurs at step "D" in the DUCK acronym.
Determine winner - this is where you'd trigger reactions to winning/losing, like Tears of Lys, Lannisport, and Ghaston Grey
Unopposed bonus - this is where you'd trigger reactions to claiming the unopposed bonus, like Rise of the Kraken
Claim - this is where you'd trigger reactions to claim, for example if a Lord or Lady is killed for military claim, Joffrey can trigger here
Keywords - finally, keywords and their reactions happen last, for example, when Robert gains a power for renown, you can trigger Lightbringer