1. A system for maintaining the privacy of a client computing device, the system comprising: one or more computer processors; a computer memory accessible by at least one of the one or more computer processors; and a network computing component comprising an executable software module in the computer memory, the executable software module executed by the one or more computer processors, wherein the network computing component is operable to: host a virtual machine instance including a network-based browsing application on the network computing component, the network-based browsing application in communication with a browsing application of a client computing device; receive, from the browsing application of the client computing device, an original browse session request for a network resource, the original browse session request comprising a network address of the network resource and identification data associated with the client computing device; in response to receiving the original browse session request for the network resource, generate an independent browse session request for the network resource, by the network-based browsing application, wherein the independent browse session request corresponds to a browse session request for the network resource identified in the original browse session request and wherein the independent browse session request does not include the identification data associated with the client computing device; cause transmission of the independent browse session request for the network resource to a network resource provider; receive a response from the network resource provider, wherein the response comprises the requested network resource and an identification token, the identification token associated with the network-based browsing application; and in response to receiving the response: determine a remote session browsing configuration associated with the network-based browsing application and the browsing application of the client computing device, wherein the remote session browsing configuration comprises a remote session communication protocol and comprises information designating respective processing actions performed on the requested network resource by either the network-based browsing application or by the browsing application of the client computing device; process, with the network-based browsing application, at least a portion of the requested network resource; and transmit a portion of the response to the browsing application of the client computing device from the network-based browsing application via the remote session communication protocol, wherein the transmitted portion of the response does not comprise the identification token.