"Keep an eye out for it" is a command. It's the owner of the dog telling the person to passively search for the dog. It essentially means "if you see my dog let me know". It's more polite to phrase it as a question though e.g. "will you keep an eye out for it?" or "will you keep an eye out for it for me?" "I'll keep an eye out for it" is you volunteering to help in a passive manner. E.g. If you met the owner on your way home from the supermarket you would passively scan the streets for his dog for the rest of your journey home paying closer attention to any dogs you see wandering the streets, and if you happened to spot the owner's dog you would contact the owner and tell him where it is. You could also grab it by the collar and wait with it until the owner arrives, so it can't wander off somewhere else. You're right. If you're keeping an eye out for something you aren't actively searching for it.

Yes Frank, you are right. When we say we will "keep an eye out for something" we are indicating that we are willing to do a Passive, rather than an Active Search. If we see something important as we go through our daily routine, we are willing to contact the right people; but we will not be abandoning our routine to Actively Search. .