Despite the loose use of language, I think the transporters must be phasing the original matter, quantum-linking the original atoms with energy that is swapped in from whatever dimension the individuals are being phased into. The 'energy' is then locked into an annular confinement beam and sent via subspace to the destination. When the confinement beam is released, the original matter naturally reverts from its dimensional state, which is how the matter can be re-assembled despite not having transporter equipment at the destination.
Some energy inevitably leaks away every time someone is beamed so when beaming back to the ship the computer compares the current scan with the buffer scan and replicates matter to replace any that is missing. This gets around a lot of transporter story issues: it prevents cloning and reparation of serious injuries (replicated cells are essentially dead matter without a live host to house them); it explains why transporter malfunctions like in TMP can kill (the quantum links get screwed up or too much energy leaks away so that you either materialise very wrongly or you can only get a largely replicated corpse back); transporter duplicates are possible only with weird malfunctions that allow a larger portion of replicated matter to still produce live beings; it is possible for genes to be filtered out by selectively deleting quantum links as long as the percentage of replicated replacement genes is not too high (good/evil Kirk or Tuvix, rascals).
It also adds in fun limitations - beaming to a series of unrelated locations over a short space of time without returning to your original destination could lead to cumulative cell damage as your 'leakage' is never repaired.
The main thing that does not make sense to me is how a ship can carry out a quantum scan from orbit to beam someone back without a localised quantum scanner. Communicators fulfill this function well but too many stories allow people to be beamed out even without communicators. It was a Pandora's box that should never have been opened.
Even stranger is the seriously long distance beaming of living matter in the NuUniverse. Even travelling in subspace is not instantaneous. It isn't clear what is maintaining the annular confinement beams at all, let alone for that long. In the comics, Scotty successfully transports a transportee's signal via standard, unmodified subspace communications relays that are only set up to transmit data. It would be nice if the modern writers could settle on some pretend physics and stick with it.