Yes, 배고파요 can mean both "I'm hungry," with a period at the end and "Are you hungry? with a question mark at the end. In fact, 배고파요. can mean he's hungry, they're hungry, etc. And, the same goes for 배고파요? i.e. (나는) 배고파요. = I'm hungry. (그는) 배고파요. = He's hungry. (우리는) 배고파요. = We're hungry. (그들은) 배고파요.= They're hungry. (당신은) 배고파요? Are you hungry? (그녀는) 배고파요? Is she hungry? (그들은) 배고파요? Are they hungry?

Since it's a question, it doesn't mean "I am hungry" and it's doubtful someone is asking themselves if they are hungry - "Am I hungry?" - since they would already know that. But... It can also mean "Is he hungry?", "Is she hungry?", "Are they hungry?". Welcome to the wonderful world of Korean!