Destiny isn't exactly peer-to-peer, where the simulation runs on every single node and results are relayed to every single node. It is a hybrid client/server model. I forget the details, I'll have to try to dig them up again, but different players' consoles are assigned different roles. Most are just clients but one or more provide server roles and act as the authoritative servers for their role in the simulation. If I can find the source I'll reply again. True peer-to-peer would be even worse.
Latency is VERY variable and is affected by CONGESTION. When a player's network is saturated, the players packets get to wait in a queue, the more congestion the higher the LATENCY. Packet loss happens most frequently when CONGESTION leads to packets being DROPPED (aka packet loss) and re-sent. There are of course other causes.
There are two things the player can do to try to mitigate packet loss -- limit latency as much as possible by using a wired connection, and trying to avoid congestion by not using bursty network apps while playing -- streaming audio and video and downloading large files.