I'm not sure they have to know your Apple ID, I would hope not, anyway. When I opened FT, it asked me to log into the Apple cloud. It then displayed a list of some of my Apple email accounts and my iPhone number. As far as I know, it's basically a video chat system that uses iCloud. If you've ever sent anything with your iCloud addy to anyone, and they also have FT, they can use the system. Nothing more nefarious than that. And, if you have ever registered a Mac, you have an iCloud account and an email address, whether you use it or not. That's my opinion, anyway. It appears that everyone in your Contacts list is accessible with FT, assuming it is compatible with whatever video chat app they might have.
So, another way to avoid these notifications might be to open FT and remove any of the "You can be reached ... at: info.
Oh, yeah, you can also turn FT "Off/On" on that screen.
I suppose all this complaining means some of us can't share computer room pictures?
BTW, here's something to do while you're bored and have a w i d e monitor. Grab the FT window and drag it from one side of the screen/monitor to the other while watching yourself. Notice that your eyes move to seemingly watch the side of the screen you are moving from and to! AMAZING! How do they do that? I'm sure those who felt the scrolling change made in Mountain Lion have complained about that, already!
I may try turning my monitor upside down and see if it still does that...
I gotta go, we're putting our Christmas tree and decorations back to 'bed' today...