Does one wall socket share power between both ac plugs?
It depends. Outlets are manufactured so that both sockets are powered from a single source - one hot wire and one neutral wire and a ground wire for safety. 

On almost all outlets, those sockets are connected by small tabs of brass that feeds the connections from one socket to the other. If you attach the hot and neutral wires to the screws for one socket, these brass tabs carry that connection to the other socket. It works the same way if you use the press in connections which are holes on the back of some outlets (but many pros and DIYers do not favor using press in connections). The brass tabs on the side carry the power from one socket to the other.

You can change this.  If you break off the brass tab on the hot side, the hot wire you attach to one socket does NOT carry over to the other socket within the outlet. This technique is often used to have an outlet with an always live socket (for clocks, etc,) and a switched outlet (for bedside lamps). The outlet is wired with two separate hot leads, one that is always hot and one that goes through a switch.  They could be on the same or different breakers depending on several factors. The neutral side may or may not be connected depending on several factors that are not critical here.

You can also separate the sockets if you wanted a dedicated socket for a high draw device, such as a heater, and run a separate line from the panel. (This generally would require a separate neutral).

In your example, if the sockets are set up conventionally, that is powered from a single line, there is no difference, except plugging into a socket is a marginally safer approach than multiple splitters.  Be sure that the overall load that is likely to be used at one time does not exceed the capacity of the surge protectors or the circuit breaker on the line.