In the case of distributing general monster loot or opening dungeon end-chests, this principle means that each player gets their own roll, so it’s alright if you are soloing and someone begins fighting alongside you. This won’t cause the loot you would receive to degrade in any way, as long as you actively participate in that combat. Likewise, when you get to the end of that big dungeon with your group, you each get to individually open the chest and receive your own personal reward.
In the case of gathering materials from things like ore nodes, plants, and the like, this means that when you gather from that resource you use it up for yourself, but not for others. In Guild Wars 2 there is no need to race to beat other people to the same resource node. Take your time ripping that bear’s head off, because no one can walk up and steal that copper node in the back of its cave from you. You may be helping others in your world reach that copper safely, but rest assured that you’re not just clearing a path for a node ganker.