Our regular licenses are activated per seat. The activation is permanent, so it will work without Internet connection as well.
If you have specifically requested to switch your license to a floating license, it will require continuous Internet connection. It will update the session state once every several minutes. If our server is unreachable during 2 or more attempts to update the key, VisualGDB will stop working and you would need to restart Visual Studio. Again, this only applies if you have specifically requested a floating license.
According to our records, the license associated with your email address is not a floating license. Hence it uses permanent activation and will not require Internet connection afterwards.