Suppose the transporter beams down two copies by mistake. Person P exists on the planet after being beamed down. Person Q is a mile away from person P, also having just beamed down. Persons P and Q will both claim to be the person R who existed before the split, and both have an equally flawless claim to it. Yet it's clear that no one is going to be conscious of what both of the pairs of eyes a mile apart are seeing, so there are undoubtedly two distinct conscious beings. This means that if we're to consider person R to still exist, they can't both be person R since they're distinct individuals. Yet if person X exists back on the transporter platform never having been destroyed, we can't really justify person X as having any better a claim to being person R than persons P and Q have. (The only unique feature of X is four dimensional continutity with R, but it's not as though we say a person is the big bang because they have four dimensional continuity with that [or a part of that, if we could speak of parts there].)
It seems we just can't find person R. We know person R was a conscious individual, and we know that P, Q and X are conscious individuals which have properties in common with R, but all we've got is a bunch of distinct individuals who we can properly say all used to be R.
Perhaps there's just nothing to the concept of personal identity over time anyhow, other than memory creating an illusion of it. Not a comfortable thought, but it's hard to rule it out. This would mean that person R simply doesn't exist for more than an indefinable moment anyhow, regardless of the transporter.