Betweenness centrality is a fundamental centrality measure that quantifies how important a node or an edge is, within a network, based on how often it lies on the shortest paths between all pairs of nodes. In this paper, we develop a scalable distributed algorithm, which enables every node in a network to estimate its own betweenness and the betweenness of edges incident on it with only local interaction and without any centralized coordination, nor high memory usages. The development is based on exploiting various local properties of shortest paths, and on formulating and solving an unconstrained distributed optimization problem. We also evaluate the algorithm performance via simulation on a number of random geometric graphs, showing that it yields betweenness estimates that are fairly accurate in terms of ordering.