I have always believed in the destroyed original/new facsimile version.
The reasons I’ve heard are as follows.
Your memories/personality are stored in the brain as physical/chemical values that can be recreated.
We do this ourselves slowly IRL. Biologist say we replace every atom/cell in our body every 7-11 years.
Thomas Riker could not exist as he does if the transporter did not kill the user. If both Thomas and William Riker were continuities of the same consciousness, William Riker would know they left Thomas Riker behind, just as easily as he knew whether his left hand was holding up two fingers or three.
Huh?
So when an embryo splits and becomes twins the original embryo is killed? Or else all twins would always be experiencing the same thing as the other?
Continuities of the same consciousness is not the same thing as being the same consciousness.
There’s a difference between Star Trek canon and the philosophical argument over bodily reconstitution. The first is a cite, the second is an argument.
I’m squarely in the “you die” camp. The pro-cloning = you teleport people must either hate or not grok the conflict in the standard clone episode that every show seems to do, especially cartoons. Like why the original wants to not be replaced so badly, or why the duplicates don’t want to thrown away like they’re garbage.
It would be interesting if your consciousness was a sort of nebulous hivemind that could be sent into multiple bodies if the targets had the same or similar hardware to receive the consciousness. You see that in sci-fi aliens or robots a lot. It’s weird to imagine, since coordinating one body can be tough enough. But that’s the only way I can see disintegration not leading to a cessation of my current consciousness, since then my consciousness wouldn’t be tied to my brain and would be kinda floating out there. That way 50 clones could all be my unique experiential self, instead of 50 duplicates having 50 unique experiences. But for humans I don’t see any reason to think it’s not one seat of consciousness, one body. Well, split brain experiments aside, but that’s a different topic.
Same thing when sci-fi/fantasy villains have backup copies of themselves as a backup plan for when they die. That guy who died stopped having experiences, his unique self doesn’t get to see if his diabolical plans work or not. The only immortal is whoever’s the latest copy to wake up. Unless the guy has a magic soul that’s jumping from body to body and he just keeps making empty vessels for backups, as is sometimes the case.
Obviously mass disappears off the transporter platform and appears on the planet (or vice-versa). So I vote for “physically transported”. BTW, I don’t think the transporter works by classically disassembling atoms; it’s probably more like they’re converted into an intangible quantum wave function which is recollapsed at its destination.
When an embryo splits and becomes twins there is no consciousness to continue. It occurs before the earliest precursor of the nervous system even begins to form.
That would seem to be an argument against anyone having a continuous consciousness…
Don’t forget the Well of Souls!
So what? By definition, atoms are identical to other molecules of the same element. If you have some sort of molecular-level replicator that can reproduce a steak, it doesn’t matter if it never came from a cow.
A cow is just a reactor that converts grass molecules into energy and delicious meat.
You would have to do it instantly, and reassemble you in a way that doesn’t damage your organs, cell structure or DNA.
Your example isn’t a good metaphor. You’re talking about chopping up a person at a macro level, which is different from doing it at the molecular level.
Personally, I think a better argument is one related to the sheer volume of information you would need to store to reproduce a person on a molecular level. I mean how much computing power is needed to decode a single molecule of DNA?
… Which is why they have Heisenberg compensators! 
This, to me, is the key point. The transporter is only doing what nature already does to our bodies, only much much faster. But that is only a difference of degree, not a difference of kind.
By the by, I have a Papermate ballpoint pen that I’ve had since 1973. I’ve replaced the cartridge many times, and the spring a few times, and the top half of the body about three times, and the bottom half of the body twice. Is it the same pen? I always claim that it is. I call it “The Pen of Theseus.”
Bingo.
FTW.
Since folks aren’t bothering to read the other thread, I’ll summarize my view: continuity of consciousness is an illusion caused by memory anyway, so there’s no significant difference between going unconscious and regaining consciousness, versus being disassembled and reassembled. Clearly, it’s not the atoms that matter, so I don’t care whether the atoms are transmitted or just the information.
I’d step into a transporter without a moment’s hesitation. The result, Learjeff2, wouldn’t either. 
(Actually, he’s Learjeff20440 anyway.)
Yeah, it’s a little long to wade through.
In any case, this thread isn’t about the existential aspects of teleportation. It’s not about how a transporter works, nor how it possibly could work. It’s about this: According to The Official Star Trek Writer’s Guide, Gene Roddenberry’s notes, and other official/approved sources, is the individual or object at the destination the same individual or object that left the origin and not a facsimile thereof?
Let me put it another way. Suppose I have a device called a ‘door’. I step through the device and I find myself outside of my house. When I get outside, am I the same person who went through the portal from inside of my house? Or did this ‘door’ destroy me and create an exact facsimile without my noticing? It doesn’t matter how it works; just the end result. Where it gets sticky, and where people start answering questions that were not asked, is when you consider ‘Well, if the result is this then how does it work? And if it works that way, then how can you possibly still be you?’ It’s not important. We all know that the reason there are transporters at all, is because it was too expensive to build and film models for practical effects. What I’m after is the intention of Roddenberry et. al. aside from the practical aspects of film production, as to what comes out the other end.
Based on answers I quoted from responders upthread, I’m concluding that Roddenberry’s intention was that transporters teleport people and objects ‘intact’, in the same way that ‘Jeannie’ or ‘Samantha’ teleported people on their shows; that transportees are not destroyed and facsimiles made of them. They step onto the transporter pad (or, in the case of objects, are placed there), and magically appear at the destination. The only difference is that you have a film set instead of a genie or a witch or one of these ‘door’ things. Except when the plot requires duplicates or multiple copies.
Lots of Roddenberry’s ideas evolved thru the years. And after he was pushed aside, they evolved even more.
I see the transporter in the same light as warp tech and subspace communications. A technological magic to make the TV stories move quickly (and cheaply).
Somewhere in Trek-dom (maybe as far back as McCoy leaving his communicator behind in A Piece of the Action), it is strongly implied, if not stated outright, that replicators, warp, transporters, and subspace communication are all part of the same leap forward in technology. Based on tri-lithium somehow. If I can find my old Trek bibles, I may find a cite…
[[last lines]
Capt. Kirk: All right, Bones, in the language of the planet, “What’s your beef?”
Dr. McCoy: Well, I don’t know how serious this is, Jim. And I don’t quite know how to tell you…
Capt. Kirk: Go ahead.
Dr. McCoy: But in all the confusion, I…
Capt. Kirk: Tell me.
Dr. McCoy: I think I left it in Bela’s office.
Capt. Kirk: You left it?
Dr. McCoy: Somewhere, I’m-I’m not certain.
Capt. Kirk: You’re not certain of what?
Dr. McCoy: I left my communicator.
Capt. Kirk: In Bela’s office?
Spock: Captain, if the Iotians, who are very bright an imitative people, should take that communicator apart…
Capt. Kirk: They will, they will. And they’ll find out how the transtator works.
Spock: The transtator is the basis for every important piece of equipment that we have - the transporter, the…
Capt. Kirk: [overlapping] Everything, everything.
Dr. McCoy: You really think it’s that serious?
Capt. Kirk: Serious? Serious, Bones? It upsets the whole percentage.
Dr. McCoy: How do you mean?
Capt. Kirk: Well, in a few years, the Iotians may demand a piece of OUR action.]("Star Trek" A Piece of the Action (TV Episode 1968) - Trivia - IMDb)
So what? “Alive” and “dead” are human concepts with no objective reality. A house and a person are both just accumulations of matter of varying complexity. There’s nothing special about a human that says you can’t break him down and reassemble him, just like you can a house. There’s no “life particle” that exists in the human and not in the house.
Sure, it kills you. Then it brings you back to life. We could do this today with a bucket of water and someone who knows CPR. Of course, we generally don’t consider that “death.” But we used to. Time was, if someone wasn’t breathing, that was it. He was dead. Then we found ways to get the lungs working again, and we redefined death as “when the heart stops.” Nowadays, we can cut someone open, rip the heart out of their chest, and stick someone else’s in there. So we moved death back again, to “when brain activity stops.” How much longer until we figure out a way to push the boundaries back again? How much further will we have pushed them by the time Star Trek is meant to happen?
Not much info on the transtator.
If we knew more about the transtator, maybe we would know the answer to this recurring question.
I’d say 100 quadrillion is much preferred to 2. I’d rather be missing 1 cell rather than missing a whole mess of cells.
Plus, if you’re chopping me in two, moving me, and then reattaching me instantaneously, I don’t think I would die just the same as if my heart stopped beating for just one nanosecond. It’d be a different matter if the teleporter worked with hacksaws and gunnysacks though.
More transtator clues.
Seriously, find out how this fictional tech is supposed to work and we probably have an answer to kill I/no kill I transporting. Anybody find more?
The canon answer is that “you” exist before and after to the same degree as if you walked through a door. Those of the Star Trek universe accepted that it was them on each side of the process even if the “original” was eliminated and the version that reappeared was a freshly assembled exact copy. (IOW LearJeff would fit in well.) Neither Riker was a copy anymore than both were and each had within their past the same experience (and consiousness) up to the transporter event and divergent experiences from then on.