When you ask for your private and public keys for the reCAPTCHA at Google’s Labs, you must choose if those keys will be used for one url only or if they are activated for multiple urls. The latter will be your choice if you are installing Skt NURCaptcha in a Multisite platform.
Activation is done by the superadmin once and for all child sites. No need of new keys even if you create other sites after the plugin activation. This feature prevents any problems due to the fact that the users database table is the same for all installations. In fact it makes no sense to leave any of the child sites opened if we want to prevent bots attacks on the database, not just on a particular url of your cluster.
To this moment the log routine creates just a .txt file inside the plugin’s directory in which occurrences are registered, one line each. No database tables at all. The idea of logging plugin activity for each site of the Network, however is a worthy wondering hint. It has been added to my agenda. Thank You.