Or does it mean both? If so, is there any way to explicitly distinguish between "I'm scared" and "I'm scary"?
If it only means one and not the other, how would you say the other?
To give a few working examples:
I'm scared of cats.
Emiko seems to be fine around cats, but apparently is scared of me.
I'm scary, just so you know.
That makes 私は怖い officially ambiguous and the answer to your question must be "both, depending on the context".