1. A computer-implemented method for managing communications for computer networks, the method comprising: receiving, by one or more computing systems of a configurable network service, configuration information specifying a range of virtual network addresses for use with a first virtual computer network that is overlaid on a substrate network, the configuration information including an indication of one or more additional computing nodes to incorporate within the first virtual computer network that are part of a remote network associated with a client of the configurable network service; selecting, by the one or more computing systems and for use in implementing the first virtual computer network for the client, multiple computing nodes from a plurality of computing nodes provided by the configurable network service for use by clients; and managing, by the one or more computing systems, operations of the first virtual computer network, including: assigning, by the one or more computing systems and to each of the additional computing nodes, a distinct virtual network address in the specified range in accordance with the received configuration information; assigning, by the one or more computing systems, at least one computing node from the multiple computing nodes to represent virtual network addresses within the specified range that are assigned to the additional computing nodes; and for each of one or more communications sent as part of the first virtual computer network to a destination virtual network address within the specified range that is assigned to one of the indicated additional computing nodes, forwarding the communication over the substrate network to the at least one computing node for further forwarding to a destination one of the additional computing nodes based on the destination virtual network address.