A computer-implemented method for providing local network access to remote network services, the method comprising the steps of: receiving, by a configurable network service and from a first customer of the configurable network service, one or more requests to initiate creation for the first customer of a first private computer network that includes multiple computing nodes (164, 165, 166) and to specify configuration information for the first private computer network, the configuration information including multiple specified network addresses for use with the first private computer network and including an indication of a network-accessible remote resource service that provides computing-related resources and that is external to the first private computer network and including network access constraint information to restrict the multiple computing nodes of the first private computer network from accessing computing systems that are not part of the first private computer network; automatically configuring, by the configurable network service, the first private computer network in accordance with the configuration information, the configuring including associating each of the multiple computing nodes with one of the specified network addresses, the configuring further including establishing an access mechanism that is part of the first private computer network and that is associated with the indicated remote resource service and assigning a network address of the specified network addresses to the access mechanism to represent the indicated remote resource service within the first private computer network, and the configuring further including performing configurations to prevent communications from the multiple computing nodes to network addresses that are not part of the specified network addresses, wherein the multiple computing nodes are a subset of a plurality of computing nodes provided by the configurable network service; and managing, by the configurable network service, communications sent by the multiple computing nodes in accordance with the configuration information, the managing including, for one or more communications sent to the assigned network address by one or more of the multiple computing nodes, forwarding the one or more communications to the remote resource service via one or more networks external to the first private computer network. receiving, by a configurable network service and from a first customer of the configurable network service, one or more requests to initiate creation for the first customer of a first private computer network that includes multiple computing nodes (164, 165, 166) and to specify configuration information for the first private computer network, the configuration information including multiple specified network addresses for use with the first private computer network and including an indication of a network-accessible remote resource service that provides computing-related resources and that is external to the first private computer network and including network access constraint information to restrict the multiple computing nodes of the first private computer network from accessing computing systems that are not part of the first private computer network; automatically configuring, by the configurable network service, the first private computer network in accordance with the configuration information, the configuring including associating each of the multiple computing nodes with one of the specified network addresses, the configuring further including establishing an access mechanism that is part of the first private computer network and that is associated with the indicated remote resource service and assigning a network address of the specified network addresses to the access mechanism to represent the indicated remote resource service within the first private computer network, and the configuring further including performing configurations to prevent communications from the multiple computing nodes to network addresses that are not part of the specified network addresses, wherein the multiple computing nodes are a subset of a plurality of computing nodes provided by the configurable network service; and managing, by the configurable network service, communications sent by the multiple computing nodes in accordance with the configuration information, the managing including, for one or more communications sent to the assigned network address by one or more of the multiple computing nodes, forwarding the one or more communications to the remote resource service via one or more networks external to the first private computer network.