This paper addresses the problem of processing range monitoring queries, each of which continuously retrieves moving objects that are currently located within a given query range. In particular, this paper focuses on processing range monitoring queries in the road network, where movements of the objects are constrained by a predefined set of paths. One of the most important challenges of processing range monitoring queries is how to minimize the wireless communication cost and the server computation cost, both of which are heavily dependent on the amount of location-update stream generated by moving objects. The traditional centralized methods for range monitoring queries assume that moving objects periodically send location-updates to the server. However, when the number of moving objects becomes increasingly large, such an assumption may no longer be acceptable because the amount of location-update stream becomes enormous. Recently, some distributed methods have been proposed, where moving objects utilize their available computational capabilities for sending location-updates to the server only when necessary. Unfortunately, the existing distributed methods only deal with the objects moving in Euclidean space, and thus they cannot be extended to processing range monitoring queries over the objects moving along the road network. In this paper, we propose the distributed method for processing range monitoring queries in the road network. To utilize the computational capabilities of moving objects, we introduce the concept of vicinity region. A vicinity region, assigned to each moving object o, makes o monitor whether or not it should be included in the results of nearby queries. The proposed method includes (i) a new spatial index structure, called the Segment-based Space Partitioning tree (SSP-tree) whose role is to efficiently search the appropriate vicinity regions for moving objects based on their heterogeneous computational capabilities and (ii) the details of the communication strategy between the server and moving objects, which significantly reduce the wireless communication cost as well as the server computation cost. Through simulations, we verify the effectiveness for processing range monitoring queries over a large number of moving objects (up to 100,000) in the road network (modeled as an undirected graph).