I did find the PvP laggy also, though I have found Titanfall and Ghosts laggy also. I think the last game I played that I didn't feel laggy was Black OPs 2.
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I upgraded my internet connection by more than double and it still doesn't help against most crappy connected players, worst on destiny was shooting dude with tank and they didn't even loose shields and just stood there ( I was the one in the tank btw)
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I don't know if Destiny uses client side or server side hit detection or some sort of hybrid system. Client side slightly favors laggy players, server side favors low ping players.