OK I'm not even a Trekkie but I was doing some elecromag homework and I have a really cool theory on this:

The teleporter thingy actually acts more like a guitar pickup, in a more E=mc^2 type of way, entirely perfectly converting the person into energy - not matter. (This would require an analog encoding from matter to energy). The biggest difference is the pickup totally uses up the entire person, so like if you strum a guitar and it converts to a perfect electrical wave (but the guitar goes mute).

This energy is a lot easier to transfer than just matter, but the person encoded within it still only exists once in that energy. (for the guitar analogy a speaker at the other end that picks up the guitar wave, and turns it back into sound)

Its then entirely used up to power the 'person builder' in an analog way, much more accurately than were able to recreate digitally (aka why tape record are the truest form of music recording we have, it accutate to a way smaller scale than we can capture digitally.)

This would then mean that we can't just duplicate the creation process, cause the energy only flowed into the machine one time in that exact fashion, and duplicating it would require knowledge of every single atom in a person; then a way to accurately recreate that energy waveform to power the machine.

This also opens the possibility of the transporter 'missing' if somehow they moved faster than the speed of light, while the person was still being transported, and them being just a flash of light endlessly propagating throughout the void.

Idk if the things have range in the series, but it could also be that the angle a transporter can accurately capture that energy is limited, and so really far away things are too large to be able to accurately capture (unless you have a massive radar dish or something alike)