If some one placed a lure module on a PokeStop and I go there am I stealing their Pokemons? For example when I see a Pidgey spawn does that mean everyone seea the Pidgey spawn and the first who clicks wins? Because I really don't want to steal other people's Pokemon when some one paid for that.
You are not stealing other people's pokemon. You wont grab pokemon from someone's inventory (there are no Team Rocket yet).
Also, the pokemons you see on the map are there for everyone, so if you capture it, other people can still capture it without any problem. It might despawn faster but I would need confirmation on that.