There's a difference between being eager and being a brown-noser. Don't ever try to suck up to people, but do try to do a good job. Yes, you might work with people whose work you admire, whose steps you follow on Twitter and Instagram, but keep in mind you're here to put in work, and they got here by doing that too. Prove yourself with your work ethic, not your compliments. And if you finish a task, don't hesitate to ask for another one—the worst thing you could do at an internship is look bored. There's always something going on at a magazine, so don't be afraid to ask who needs a hand.
Be proactive without being annoying.