I've taken a stab at this on a previous thread:
"Realm of Fear contradicts a lot of the pre-existing language relating to transporters but comes closer to explaining how transporters must work to avoid killing transportees IMO.
There must be a dimension outside time (per Relics) and space to which the transporter has access. The system scans and quantum links every atom in your body with energy from this dimension and the transporter switches those atoms with their quantum linked counterparts (not exactly quantum linking as we currently know it but some kind of futuristic equivalent). So the transporter's own power keeps the person in a dimensional pocket and the linked energy is kept in our dimension by the confinement beam. The energy then has to be transmitted to the desired location and when the confinement beam is removed the energy and the person snap back to where they should be but in the new location in our dimension.
Where I disagree is the allegation that transporting does no DNA damage. It absolutely should. No confinement beam is going to be 100% perfect and some of your information is going to leak away in transit - we have seen this to be a danger in many TNG era episodes. I personally think that replicators cannot replicate living material so no simple cloning machines will exist on starships. The reason for keeping a blueprint of the transportee in the pattern buffer is so that when you beam them back to the ship you can replicate a portion of their DNA that has been lost in the two transports. If the % of the pattern that has leaked away is too high then you end up with a poorly or dead person. It also explains the term, 'Boost your matter gain,' used by Rand, I think in TMP. If the links to the information get scrambled like in TMP you get a dead mess returning when the transportee snaps back in the wrong order.
This also covers things like good/evil Kirk or Thomas Riker where the transporter was somehow able to add 50%+ replicated material to the transportee and still have them live through outside intervention.
It also means that long distance transport is possible but only if you have a way to maintain a confinement beam to your destination without leaking too much information. My issue with NuTrek transporters is largely that I can't see how the long distance transports are supposed to work using standard 23rd century technology. A new algorithm can tell you mathematically what you need to do, I just don't see how it lets you maintain a confinement beam over such long distances let alone deposit you on a moving target 'safely' in the time it takes the transport to resolve.
The degree to which one remains conscious and/or able to interact while in the pocket dimension is open to debate. TWoK and Realm of Fear suggest that there is a threshold at which you can interact while being only partly in our dimension. I don't believe that Saavik was talking to Kirk in the transport beam, she was just continuing her conversation once she materialised sufficiently to allow her brain to continue performing the task it had started while in our dimension prior to dematerialising into the pocket dimension.
This system has a few nice story elements that could have kept Trek transporting reasonably elegant. It means one cannot transport too often since you run the risk of DNA damage from too much replicated matter building up in your system. It means you need a localised quantum scanner so no more beaming up without a communication device or locking on to enemies unless you have first 'tagged' them somehow. It means emergency transporters are less accurate and sacrifice a certain level of detail in order to get a lot of people off the ship safely, which has health implications."
So in answer to the thread title, a transporter cannot exactly transport itself unless there is a separate device that remains behind to maintain a confinement beam to the destination. The dimension is outside time and space. The issue is sending enough of the quantum-linked energy to the desired destination for enough of them to snap back in the correct order.
Of course if energy is being swapped for matter it doesn't really get over the matter to energy conversion ratio. Could halve the amount of energy required by using quantum computing?
It's not how transporters were ever really described but it's how I think they should have worked.