Funny, I also had this exact discussion with myself after seeing a TNG-episode where Riker has transported from a planet to the ship, but the teleportation "beam" reflected and somehow he ended up in both places.
Now. If I were Riker, and the teleporter is removing particles from me and transmitting their info (or the particles themselves) I would die and a copy would show up at the ship. Everyone would think it was me but I won't be there, I'm in the void where many think we go when we die. Why do I think that this is the case?
Well. If it were so that my continuous experience could be transferred and I would show up at the ship, then I would Also have to show up at the planet, And I would be conscious of both bodies at the same time. I would have to be in both places at once because if the subjective unique experience can be transported then there is nothing that can really say whether I would show up in one place or the other, I would Have to end up in both. If this is in violation to the laws of nature then we have to assume that it is not possible to transport the subjective unique experience. There are theories in quantum mechanics that points to that our subjective unique experience may come from somewhere else, and that our brains are merely transceivers.
However, if our subjective and unique experience is limited to the brain, then a transport would destroy it, as the transport in star trek seemingly destroys the original body, or the atoms of it. It doesn't really matter if it's done gradually.
But basically, if we disregard the quantum theories then it's simply illogical that we can be present and have the same subjective unique experiences at two places at once. If that were the case, then they would have to converse with each other through wi-fi or something, or the subjective unique experience would not come from the brain alone. If it was outside and transmitting into the brain then perhaps it could be possible to have a clone and be present in both bodies at once. It would be like having 4 legs and 4 arms, 2 heads and all that, but in different places. Ultimately one would get used to it, but that would make you sort of a commander of 2 husks, and you could perhaps choose which body to interact with.. I donno. But it seems pretty unreal, so I don't think that's possible.
So, then we come to the conclusion that our subjective and unique experiences are bound to the continuity of our human body, if it's taken apart, things will fade out of existence and eventually things will turn black.
So, this means that teleportation Will Kill people. It will also create the same amount of people, but it Will kill. Now, the subjective unique experience might not be able to object to that as it's now not existing, but it's still killing, and saying "It might not object" is essentially saying that it might be OK to kill people with a hatchet just because they won't object to it after they're dead. That's pretty grim.
What has puzzled me even more is what will come after the continuity ends. I know that if I'm not able to observe my thoughts and be conscious, then time goes faster. If I'm not conscious at all then time is equal to 0, i.e. I teleport into the future when I sleep. So, if I die and stop being conscious, then time will again be reduced to 0, even if it's an infinity. And before I was born, time must also have been reduced to 0 in length. Will all be black? But, is that really what unconscious is? I mean, because we're not able to observe our thoughts or anything during non-REM sleep then it wouldn't be black, it would just be nothing, as time is reduced to 0 there is really nothing there, not even blackness. So one wonders what happens, when the infinity of time of not being conscious is reduced to 0. If an infinity Can be reduced to 0 that is. But perhaps my view of time is far from reality. It's somewhat sad that the ultimate mystery will only be revealed at the moment of discontinuity, and perhaps we won't even be able to contemplate on it when it happens.
That pisses me off a little.