Suppose there is a popular system that is widely used by a huge amount of people. Its security protocol provides a finite group with a generator $g$, and users need to choose a random number $r$ and calculate $h=g^r$ in a certain phase. This is commonly seen in many schemes.
Since this system is widely used by many people, the possibility of two users choosing the same $r$ may not be negligible. In that case, a user may accidentally find the random number used by another person and cause some security problems.
Is this problem worth being considered? If so, is there any way to prevent it? Or, am I wrong that the probability of finding such a collision is actually negligible?