Imagine that instant teleportation is possible. Take the simplest scenario, some guy is at rest in our frame until he suddenly disappears and reappears X miles away at the same time. Let us forget for a moment about how everyone will see him at two places at the same time, and focus on how he will experience the jump. So far I have done the easy part of this:
This picture is about when he thinks events happen around him. This is done in old good way - shooting light beams at things at t1, receiving them back at t2, and concluding that things reflected the beam at (t1 + t2) / 2. This is what our guy would do on any other day when he is at rest and, if he was not informed about being subject to secret government experiment, this is what he surely will do now.
Let's skip over the boring parts: green is when everything is normal, red and blue are time shifts (where observer at rest would see events before or after our poor guy), and the yellow zone is where everything happens twice - things are barely unexpected for now. But if we look at what happens at grey areas - it is far more interesting.
I named dark grey area Vegas, because what happens in Vegas stays in Vegas. This area is "black hole" -like, whatever signal you send there, or someone sends from there, cannot reach our guy. These events basically do not exist as far as he is concerned.
Finally, the light grey area is the most interesting. Events that happen there do affect our guy. For example, if he did teleport in order to avoid nuclear explosion that was scheduled in his area, he would see it happened. BUT - he has no way of telling when did it happen. This part is blowing my mind the most.
I mean, he could, in a good faith, attempt to extrapolate iso-time lines from other regions there, but there would be no physical reason to chose either green or blue are to do so, nor to conclude that these events happened twice at the different times - the notion of time for events in that area seems to be completely undefined. As such I feel that I am missing something here.
Help? Just don't call the doctor yet.
this could be interpreted as if all the events on the way happen at the same time to our guy, and then it does no longer look that paradoxal.
)) - however, his earlier self has no chance of knowing that his later self will be doing it again, so there is no paradox (edit: to have paradox you probably need to allow him to reappear in the bottom green area - but that is no longer teleportation but full-scale "let me have the cake and eat it too" time travelling).
What would they want with me anyway? 
Naturally the new you would think it is you! 
