From Destinypedia, the Destiny wiki
Do you know what the player on the other side meant? It was an idea that people talked about long ago, before the Golden Age. And the idea was that each of us has a player on the other side—a perfect enemy, just as powerful, just as smart, just as experienced. And that perfect enemy was waiting for each of us, waiting patiently, because the day of meeting would come in time. The other-side enemy didn't have to rush it. Sheer time and chance would bring you together.
Think of that. Right now, someone out there is waiting for you, and they know you so well they practically are you. That's the player on the other side.
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