This paper studies problems on locally stopping distributed consensus algorithms over networks where each node updates its state by interacting with its neighbors and decides by itself whether certain level of agreement has been achieved among nodes. Since an individual node is unable to access the states of those beyond its neighbors, this problem becomes challenging. In this work, we first define the stopping problem for generic distributed algorithms. Then, a distributed algorithm is explicitly provided for each node to stop consensus updating by exploring the relationship between the so-called local and global consensus. Finally, we show both in theory and simulation that its effectiveness depends both on the network size and the structure.