Before I give my opinion about this question, I first need to provide a disclaimer. I'm not all that into astrology. I don't know why. Maybe it's my scientific bent, or the fact that I didn't grow up with it; still, it's just not my thing. Now you know.
Still, the question of whether people born under the sign of Gemini are naturally empathic is an interesting one. I mean, it makes sense, since Gemini people are under the sign of twins who were torn apart and then reunited. It would seem that the surviving twin would naturally have their third-eye searchlight on.
But let's go back to the original myth of the Gemini twins. In the Greek version, the Gemini twins were actually two sets: Castor and Pollux were boys and Helen (later of Troy) and Clytaemnestra were girls. Both Pollux and Helen were children of Zeus and were demigods. Of course, we usually hear of the boy twins and how they were inseparable. When Castor was killed in battle, Pollux mourned deeply. Pollux asked his father Zeus to return his brother to him. Since this was not possible, Pollux ascended into the heavens to join his brother and ever since then the twins have been together.
According to the wonderful book, Mythic Astrology: Internalizing the Planetary Powers by Ariel Guttman and Kenneth Johnson, this myth not only symbolizes the need to balance the male and female aspects of our personality but also the mortal and eternal. The book goes on to describe how the Gemini constellation appears to other cultures and the lessons to be learned there. If you're interested, please refer to the book. It's a good one.
But are Geminis naturally empathic? More to come.