on the watch (for someone or something)
Actively and attentively looking out for someone or something. You need to be on the watch for suspicious people when you're out in the city this late at night. A: "We've had reports of counterfeit products coming through the shipping yard." B: "Don't worry, I've already put extra inspectors on the watch."
on the watch (for someone or something)
alert and watching for someone or something. Please stay on the watch for trouble. I'm always on the watch for Ann. I want to know when she's around.