First I'd also had the same feeling that if someone is working on a mine or a tree, I should leave him alone, but after a bit of observation, I found that's not completely true, here's my thought:
1. There is a maximum number of use for each mining point, when it reaches to the limit, the point would disappear.
2. You can only mine a certain number of times per point, as your rank goes up, this number goes up.
3. When you mine at a mining point for X times and even if the point is still there, you can not mine from that point anymore. It'll say something like "you can't mine at this location anymore".
So let's say a mining point can be mined for 10 times, and say at your current rank, you can only mine 4 times, so by the time you're done, the mine would still be there and if no one access it, it seems it'll stay there forever (I'd never seen a mining point disappear without anyone using it) and you can't mine on it anymore. So if there's another person sharing the point with you, and it happens that he can mine 7 times, so if you started mining before him, you'll most likely finish mining it 4 times and after he mined for his 6th, the mine will disappear.
I'm assuming these because these had happened to me:
- I saw a mining point for distance and someone was working on it, so I was just standing next to him and watching him. By the time he's done (put his pickaxe away), the point is still flashing, so I started mining and when I'm done with it, the point was still there.
- I started mining at a mining point, then someone joined me. By the time I'm done, the point disappeared, then both of us put away our pickaxe.
- I finished mining at a mining point, it's still there, so I went to another mining point, finished mining there, then came back to the first one and tried to mine it again, but it wouldn't let me.
BTW, I'm wondering at what level we can get the ability to show the mining points on the map? My miner was rank 8 and I still didn't have that ability. Tho I remembered when I got to rank 4 or something, it said I learned Prospect, but I couldn't find it on my ability list anywhere...