Techniques are described for providing users with access to computer networks, such as to enable users to interact with a remote configurable network service in order to create and configure computer networks that are provided by the configurable network service for use by the users. Computer networks provided by the configurable network service may be configured to be private computer networks that are accessible only by the users who create them, and may each be created and configured by a client of the configurable network service to be an extension to an existing computer network of the client, such as a private computer network extension to an existing private computer network of the client. If so, secure private access between an existing computer network and new computer network extension that is being provided may be enabled using one or more VPN connections or other private access mechanisms.