The problem you are raising here is actually very nicely discussed in Derek Parfit's famous book Reasons and Persons (Part III). Parfit asks you to imagine tele-transportation, where your body is carefully scanned (and destroyed in the process), the data e-mailed to some destination, and a human being constructed at this destination who is an exact replica of you, including your memories and whim for hazelnut chocolate. You'd be scared to travel this way, but seeing that others do it safely all the time, you too do it and get used to it.
Now one can ask whether the person getting out of the machine at the destination really is the same person as the one who walked into another machine at the departure point. As Allen Stairs wrote back then, something can be said for either answer. But there's a third thing one might say: once the story's been told, there is not further question to be answered. You can say what you like about sameness, the important thing is that you really have no serious reason to avoid using this technology -- provided it works, of course!
One day you travel by e-mail, the technology actually works a little better than usual. The scan does not actually destroy the person at the departure point. So now we have two people, just as you imagined in your question. As you write: if they both were identical with the pre-departure person, then they'd be identical with each other -- and this they surely are not, seeing that they are miles apart from one another and having a heated conversation with each other on the telephone (about who gets to be with hubby and the kids).
When you used the technology in the old days, when the scanner destroyed the person at the departure point, you thought of e-mail travel as being just as good as taking the train. You may be a different person each time, strictly speaking, but why mind? Parfit examines the plausibility of this attitude for the branchline case, where you are the person who survived the scanning. He imagines that the scan did damage after all and that you are going to die rather soon. Can you be as cheerful about this, in light of the person at the destination point, as you used to be pre-scan in your previous e-mail travels? It would be hard to be cheerful like this in the face of imminent death, but Parfit makes a good case that you have reason to be. So have a look and see what you think.