Given the limited explanations in on-screen original series canon "converting matter to energy and back again", and the scientifically real understanding that mass and energy are equatable (E = m x c2), we have to assume that the transported subject is converted to its energy equivalent, "beamed" to its destination, and converted back. The notion of destroying ("killing") the original is entirely hand-waved. Somehow, the subject maintains a continuous existence throughout the transporting process despite everything that occurs; there is no "destroyed original" and "re-constructed copy".
Part of the transporter dilemma surfaces in the TNG episode "Realm of Fear". Barclay describes the process (being "deconstructed molecule by molecule" and converted to "gigaquads of data"). The episode also depicts the process from the perspective of the subject being transported. The depiction seems to suggest that the subject maintains continuity of awareness throughout the process, able to see "phased matter" in "the matter stream".
Remember that Star Trek only got transporters when the original series pilot production costs and schedule issues got in the way of shooting ship/shuttlecraft landing sequences; all the nasty implications and logical dilemmas it raised simply weren't addressed.