Hello Jmgbook, welcome to the Seven Forums.
I am not sure if I understand you correctly. One thing is for sure: There is absolutely no way you can get new mail notifications if you are not connected to Internet. Not with Windows Live Mail, not with any other mail client.
WLM notifies you of new mails by showing it on the notification area. Here's WLM icon on notification area when there's no new mail:
And here when new mail has arrived: