Well,
if you stop existing i think it is safe to say that the teleported person is not you; you will just die (or relocate to a better multiverse at best). Others might not notice, but you will
Actually, you probably wouldn't, at least not for long.
I like this answer:
It's not an experiment we've run yet. You can migrate your consciousness around your brain. We've not yet created artificial neuronal structures and then migrated our consciousness into and out of those structures. My suspicion is that it's possible, but I don't know for sure. I'd also like the technologies to be done by people who think it's possible.
People do indeed change out their atoms extensively over time, so in principle our recollected selves of a decade ago are dead, per how the OP posits the question.
That said, how is the teleported copy any more or less you than you?
You are presumably limiting "where" as referring to within the bounds of four dimensional objective reality. If so, what if there is a flaw in that premise? If not, there are myriad possible answers and your question poses no particular problem.
This actually posits something falsifiable within our reality if it has any relevance to the experiment, and in principle could be tested and confirmed/falsified as a result. Otherwise if the external influence just accounts for whatever we do while hiding from detection all the time it's an extra detail that doesn't make meaningful predictions.
People, even non-religious people, HAVE posited extra dimensions though. Not testable versions of those ideas yet, to my knowledge.
If the consciousness did not transfer, then the new body would be a blank state without any experiences except newly coming into being. Most cloned hypothesis lean toward a similiar but different and unique individual.
The copied person would in principle have identical memories, experiences, personality, and abilities actually. However, if the original wasn't destroyed and they both went on living they would diverge based on different experiences/interactions. Not just mentally, even physically.
Which one is the "real" one? I'm not sure that question is meaningful.