I guess a big question is whether a person can be copied in a non-quantum sense.
If quantum state must be copied to preserve consciousness, then there can be only one of a person. This would be most convenient, since it simplifies the question.
But, if the quantum state doesn't matter, or is only important in a transient sense, then you could copy a person.
But also, if the quantum state doesn't matter, then continuity can be questioned without considering teleportation: The person that I am today may not be continuous with the person that I was yesterday.
Brings up a lot of messy questions: If I store my physical state sufficient to recreate myself, then destroy my physical state, am I alive or dead? (We have this problem in D&D when a person is petrified. What happens to their soul?)
Or, if quantum state matters, but only in a transient sense that it is regenerated on a regular interval when consciousness arises, say, in the morning after sleep, then we might see subtle differences between instances of a person which were created from the same physical base. The person on the far end of a teleporter could really be an evil twin.
For folks watching "Dark Matter", they are playing with this idea, with a transporter implementation which uses a biological 3D printer and copied memories to effect transportation. See episode 8.
Thx!
TomB