If the card is on your FS/FT list, your potential trade partner can put it in a trade proposal. It doesn't have to be on their want list, and it didn't have to show up in one of those want list (or for sale) notifications.
Folks, those e-mails you get are NOT trade offers. (I had someone make be a trade offer where they thought I was asking for certain cards and when I asked what they were sending for the cards they were asking for, they were surprised at me asking???) Those are simply alewrts that let you know that matches (in either direction) were recently discovered. That's ALL they are. Alerts. Nothing more. (Sorry to kind of hijack the thread there, but this sounded very much related to this.)
-- Dan --
Note: Please see my profile for more info regarding trading (section updated 10/25/2023). I have added a large portion of my inventory to the site, and currently have trading turned off (details are in my profile). I have (finally) unearthed my 2 very large boxes of Star Company minor league sets and they are available (email for details).