Each time a tree or ore deposit yields a product (branch/nugget of ore), the spawn has a chance of being depleted. Now...the point at which these CNR sources give their products occurs when a certain level of damage has been reached, and whoever lands the final blow should receive the CNR. Now, I'm really unsure as to what would happen if two people landed the last blow at about the same time. In theory, it shouldn't be depleted any faster than normal, but lag and race conditions can sometimes cause some strange stuff to happen. But, whether or not a spawn depletes on any given success is still a random chance. Given the chance for problems, I'm not sure why you would have two people mining/chopping on the same spawn when there's likely another one nearby.
One last thing to note about these types of CNR. They are now limited in yield. In most cases this doesn't matter, but for anyone who has taken the occaisional 20, 30, 40 or more pieces of CNR from a single spawn in the past...well, no longer. Trees and ores now max out at 10 nuggets/branches, with a slight possibility of 11, if you get that occasional special bonus piece. This applies to ore and tree branches only.