Many have been discussing the possibility of player-versus-player for awhile now. Among the theories exist many articles asserting PVP is right around the corner; in fact, it is not even in development yet.
During the recent exchange, John Hanke explains trading is right around the corner, but PVP is a concept that is still in the talks. Yes, you read that correctly: It has not been implemented into the game, but most likely will be. When? Very, very far from now. Below is the exchange:
IF: What about things like trading or battling your friends directly, are those things we’re likely to see, or [are they] really [not] part of the spirit of what you have in mind?
No, those are great social activities, and we’ve said in the past that trading is coming, and player-versus-player battling is something that we talk a lot about internally, and I think it very much is within the spirit of Pokémon Go. So it’s possible you’ll see that as well.