"Two people at once" isn't how Nagata puts it. In her terminology, one being, the original person, continues in standard embodied form, while another being, a "ghost" -- inhabits some other location, typically someone else's "atrium". Suppose you want to have an intimate conversation long-distance. In Nagata's world, you can do it like this: Create a duplicate of your entire psychology (memories, personality traits, etc. -- for the sake of argument, let's allow that this can be done) and transfer that information to someone else. The recipient then implements your psychology in a dedicated processing space, her atrium. At the same time, your physical appearance is overlaid upon the recipient's sensory inputs. To her (though to no one else around) it will look like you are in the room. The person hosting you in her atrium will then interact with you, for example by saying "Hi, long time no see!" Her speech will be received as inputs to the virtual ghost-you in her atrium, and this ghost-you will react in just the same way you would react, for example by saying "You haven't aged a bit!" and stepping forward for a hug. Your host will then experience that speech overlaid on her auditory inputs, your bodily movement overlaid on her visual inputs, and the warmth of your hug overlaid on her tactile inputs. She will react accordingly, and so forth.
The ghost in the atrium will, of course, consciously experience all this (no Searlean skepticism about conscious AI here). When the conversation is over, the atrium will be emptied and the full memory of these experiences will be messaged back to the original you. The original you -- which meanwhile has been having its own stream of experiences -- will accept the received memories as autobiographical. The newly re-merged you, on Earth, will remember that conversation you had on Mars, which occurred on the same day you were also busy doing lots of other things on Earth.
If you know the personal identity literature in philosophy, you might think of instantiating the ghost as a "fission" case -- a case in which one person splits into two different people, similar to the case of having each hemisphere of your brain is transplanted separately into a different body, or the case of stepping into a transporter on Earth and having copies of you emerge simultaneously on Mars and Venus to go their separate ways ever after. Philosophers usually suppose that such fissions produce two distinct identities.
The Nagata case is different. You fission, and both of the resulting fission products know they are going to merge back together again; and then once they do merge, both strands of the history are regarded equally as part of your autobiography. The merged entity regards itself as being responsible for the actions of the split-off ghost -- can be embarrassed by its gaffes, held to its promises, and prosecuted for its crimes, and it will act out the ghost's decisions without needing to rethink them.
Contrast assimilation into the Borg of the Star Trek universe. The Borg, a large group entity, absorbs the memories of various assimilated beings (like individual human beings). But the Borg treats the personal history of the assimilated being non-autobiographically -- for example without accepting responsibility for the assimilated entity's past actions and plans.
What makes the difference between an identity-preserving fission-and-merge and an identity-breaking fission-and-merge is, I propose, the entities' implicit and explicit attitudes about the merge. If pre-fission I think "I am going to be Eric Schwitzgebel, in two places", and then in the fissioned state I think "I am here but another copy of me is also running elsewhere", and then after fusion I think "Both of those Eric Schwitzgebels are equally part of my own past" -- and if I also implicitly accept all this, e.g., by not feeling compelled to rethink one Eric Schwitzgebel's decisions more than the other's -- and perhaps especially if the rest of society shares my view of these matters, then I have been one entity in two places.
To see that this is really about the content of the relevant attitudes and not about, say, the kind of continuity of memory, values, and personality usually emphasized in psychological approaches to personal identity, consider what would happen if I had a very different attitude toward ghosts. If I saw the ghost as a mere slave distinct from me, then during the split my ghost might be thinking "damn, I'm only a ghost and my life will expire at the end of this conversation"; and after the merge, I'll tend to think of my ghost's behaviors as not really having been my own, despite my memories of those behaviors from a first-person point of view. The ghost will not bothered having made decisions or promises intending to bind me, knowing I would not accept them as my own if he did. And I'll be embarrassed by the ghost's behavior not in the same way I would be embarrassed by my own behavior but instead in something like the way I would be embarrassed by a child's or employee's behavior -- especially, perhaps, if the ghost does something that I wouldn't have done in light of its knowledge that, being merely a ghost, it would imminently die. The metaphysics of identity will thus turn upon the participant beings' attitudes about what preserves identity.