Star Trek transporters and Consciousness

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The Star Trek transporter is supposed to dematerialize a person, beaming the energy to a remote location and then materializing a person again but wouldn't that kill the person in the origin and assemble a new person with the same identity in the destination? Of course that, for the person on the destination, he would still be the same person. Memory would provide him with the illusion of a continuum but for the mind that was active before the process, there would be no destination. It would be shut down as the physical structure that sustains it is shredded into pure energy and then an exact replica would be set up with the same set of particles.

As a kid, I imagined a way to assemble a transportation system like this but instead of sending the energy of a dematerialized person, it would scan all the particles in our body, send that information to another place, then it would assemble the physical structure of our body while the original was destroyed. Assuming that all of the components of the mind (like memory and personality) are preserved in the brain's physical structure, this would actually work! People would remember getting into a machine and then appearing somewhere else instantly and carrying on with their lives. There would be no practical problems with this so if I ask you if the person that was transported is still the same person you might say yes, but now let's imagine this with a twist.

What if the original set of particles (you) was not destroyed. Now you have two you's, one that never left the starting location and another in the destination. Now it becomes obvious that they're not the same person, they might have the same identification in that society but they're two persons, not one and the same. If I put myself on the shoes of the person that arrived at the destination, I would have no idea that there was another me somewhere else. I wouldn't feel his pain, I wouldn't see what he's seeing and I wouldn't experience his thoughts. It would be a third party for me, a different person that just happens to have the same memories, personality and body up to a certain point in time. It would be a clone, a replica, not me. Could I say that I "am" that other person? So now the answer seems no, contrary to the other example above, but wouldn't this mean that in the other example, where the original is destroyed, that you (the original) would die and simply cease to exist? And that a whole new person would come into existence thinking that it has always existed? Wouldn't everyone keep dying in the transporter without anyone ever noticing?

Now is the fact that the person on the destination is set up with the same particles make any difference? The first is still destroyed and a new person is assembled as in the first example. Unless what "you" are is really that exact set of particles but in that case how would we deal with the fact that the particles in our bodies are constantly being recycled. From time to time, we can be sure that no single particle in our bodies are the same so wouldn't this mean that we're not the same person? Is timing the difference? So if I change them gradually it's still me but if I change them too fast it's not me? What sense would that make?