This is a metaphor, not a common expression in English, but is an unexpected, colloquial construct used to convey a feeling. I would interpret this as "benefit from" but in a way that conveys the feeling of the one reporter sharing food as one would to a puppy. It is combining an expression like "copy off my test" (where someone would be cheating) with the image of a cute puppy. It is prefaced by "Why compete when we can work together? You can share my press cards" which suggests that instead of working against each other, they could cooperate. Adding the image of a "puppy dog" suggests they could be pals, although it also conveys a subordinate relationship.
It is common in English for nouns to be used as verbs -- in fact for any part of speech to be used as another. A word may be with a shift in grammar as part of a metaphor and then later can become a standard part of the language.
"English is amenable to functional shift because current English has relatively few forms that identify a word as belonging to a particular grammatical category... a word that enters English as a member of one grammatical class can quickly shift to another class. Historically, this is what happened with instance, loot, stucco, and trash, all of which entered English as nouns but have become verbs as well." --Slang and Sociability: In-Group Language Among College Students