I have not played this specific game, however my son tried to play some Xbox 360 online games and experienced some of the same problems. Depending on the game it may show your latency or connection quality when you "host" a session or room (not sure what the proper terminology would be for Destiny). If other players see that the latency is high (satellite is 650 ms - 900 ms in my experience) they will forgo the session to join a better quality host.
This is because of the lag issues when playing with someone with a slower connection. One of my son's games he played online caused him to rack up kills because it caused everyone to lag so badly that he had an advantage due to the way the game dealt with the lag.
We ended up canceling the 360 subscription once it ran out since it just didn't do that well for him and he goes over to friend's houses if he gets the inkling to want to play now.
Hope this helps shed a little light.
There is nothing anyone can do about the latency that hasn't been done to get it as low as it is, that's just the limitation of physics for what it takes for us to get internet service at all. Some games work fine online others fail miserably, most fall in the middle, work but not ideally so.