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No, they use zero-day exploits in common media formats. The spy sends you a message containing an image or pdf, your device parses it, is exploited, and then removes the message, before there ever is a notification about it. You will never know that it ever happened. For example, see FORCEDENTRY, which is one of theirs, and the technical deep dive of it is about the most amazing piece of technical writing released la…
I can't believe that all text messages aren't stored somewhere on the NSA (or equiv.) server (so it should be easy to quickly find the zero-day after a single attack). They probably just aren't motivated enough to expose the zero-days associated with it.
Also for iPhones it's usually iMessage instead of SMS which supposedly is e2e encrypted.