Maybe I don't understand, but doesn't a transporter located on someone's chest (and thus, part of what gets beamed) need to transport ITSELF (and thus break itself down into a matter stream and then reconstruct itself) when it beams someone from one place to another? If so, how does the process of beaming remain stable enough for the transport to happen at all?
And if so, how does it do that?
Its probably advanced to the point where a sufficiently strong signal is sent out a milisecond or microsecond before transport via subspace which is then maintained to reconstitute energy into matter at a designated transport site.
While in the 23rd and 24th centuries you needed transporter pads to lock on to an object, preserve the matter which was converted into energy and then send it to a location of your choosing that's within range, in the 32nd century, all of this is probably accomplished via the personal transporter that would likely be microscopic (or less) in size well by that point... and the only thing you need to worry about is maintaining signal cohesion... and I surmise that once you send out a signal, your energy (along with the personal transporter) are sent along this signal (via subspace) to the desired location.
Think of it like a starship using Warp or even a spore drive.
The drive and the ship transport themselves to the desired location.
The ship uses subspace to get there.
I would imagine that not very long after the 24th century, the technique for personal transporters was perfected.
We've seen in TNG that subspace transporters (while initially having a problem of leaving people with cumulative body damage), work along the same principles.
You need a simpler emitter that's the size of a wristband... and voila... so, this is probably the same technology (just perfected to eliminate any cumulative health issues) and of course minituarized to the point of being the size of a nanite or an atom (by the 32nd century, it would likely be the size of an atom... or subatomic).