Offline Messaging is only avaliable for your contacts, i.e., you can send Offline Messages only to people which you have in your contact list. When you tried to send an Offline message to a person which you don't have in your list, then "This message could not be sent..." message will appear. (I don't know why I wrote these)
I have tested it with someone who is appearing Offline, and I could send my Offline message to him and he received it.
About the Invite button: I have also tested that, and it didn't help me to detect if my contact really signed out or simply appeared Offline.
Edit: I have just noticed that this thread has been created on January 2007
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