1. A system configured to manage network communications, comprising: a processor of a computing system; and a system manager module that is configured to, when executed by the processor, manage a first network by: configuring one or more first computing devices to use supplied mapping information that includes one or more second network addresses for the one or more external second networks, wherein the one or more first computing devices are configured to communicate with the first network and with the one or more external second networks and are further configured to manage incoming communications intended for multiple computing nodes that are part of the first network based at least in part on the mapping information, each of the multiple computing nodes having a distinct first network address specified in accordance with the first network; and configuring one or more second computing devices to use the mapping information, wherein the one or more second computing devices are located within the first network and are further configured to manage outgoing communications from the first network that include a first outgoing communication sent from a source node that is one of the multiple computing nodes of the first network to an external destination that is not part of the first network, the managing of the outgoing communications by the one or more second computing devices including: determining one of the one or more second network addresses to use in the first outgoing communication for the source node based at least in part on the mapping information; forwarding the outgoing communication along with the determined one second network address to the external destination without using any network address translation functionality of the first computing devices; and initiating an update of the mapping information used by the one or more first computing devices to reflect the use of the determined one second network address for the source node.