Various episodes indicate that it's you being sent from place to place, and that you remain aware during transport. In Doomsday machine, when the transport took an unusually lengthly period of time, as soon as Scotty finished materializing he commented that he was aware that the transport was "wrong."
Yes I know this. However this isn't consistent from episode to episode. There are plenty of episodes where it is implied or stated that the transporter works more like a "scanner and printer" that scans and converts your matter into energy but while in that "energy state" you are unconscious until rematerialized.
For example in TNG "Relics" Scotty rigged the transporter to store his "pattern" and he was this way for DECADES. Wouldn't it be hell if he really was conscious all that time while in that "buffered state"? I think he would have become insane by the time he was rematerialized if he were in some conscious limbo state imprisoned in the buffer all those decades!
And again WRT "Second Chances", how exactly can both Rikers be the original consciousness? If we assume that you are conscious the whole time you are in the "energy state", then what happened when they split Riker's beam? It seems to be that when that happened (or some time afterwards) a new consciousness was formed.
Now I can't say if it was Will rather than Tom but one of the two (or both!) could not be the continuing consciousness of the original Riker!