Techniques are described for providing users with access to computer networks, such as to enable users to interact with a remote configurable network service to create and configure computer networks that are provided by the configurable network service for use by the users. Secure private access between a computer network provided for a user by the configurable network service and one or more other remote computing systems of the user (e.g., a remote private network) may be enabled in various ways. For example, a user may programmatically invoke an API provided by the configurable network service to obtain assistance in establishing remote access from a remote location to a provided computer network of the configurable network service, such as to establish a VPN connection from the remote location to the provided computer network using hardware and/or software supplied to the remote location in response to the API invocation.