It is possible to use an existing electrical outlet to install another outlet on the opposite side of the wall on which the current outlet is installed. The wires to the second outlet are threaded through the wall and connected to the electrical contacts of the existing outlet. However, this is only possible if the result will be safe. Here are some of the situations in which the result won't be safe and you shouldn't do it:
The Circuit's Breaker Regularly Trips
There are several reasons a circuit breaker can trip; the most common ones are circuit overloads and electrical malfunctions. Therefore, if the outlet you want to use for the expansion is part of a circuit whose breaker always trips, you should put your expansion plans on hold until you can diagnose the problem. For example, it would be extremely dangerous to use the outlet for an expansion if it's located in a circuit that is already overloaded. That would be an easy way of killing the circuit (and making it susceptible to electrical fire) because the breaker would be tripping continuously.
The Circuit Already Has the Maximum Number of Lights
There is a finite number of lights that can be connected to it. The maximum light fixtures an outlet can hold depends on the current rating of the circuit and the wattage (power) rating of the light fixtures. However, don't forget that you are not supposed to end up with a hundred percent match i.e. you should not give the circuit its maximum load. The National Electrical Code (NEC) recommends leaving it at 85%. Therefore, if the light fixtures in the circuit already represent 85% of its load, don't use the outlet for expansion.
The Outlet Can't Handle Additional Wires
Another thing to determine is whether the outlet can handle additional wires or not. Whether this is possible depends on the gauge (size) of the wires connected to the outlet as well as the volume of the outlet box. For example, if the box is small or the wires are big, then it's possible that the outlet should not hold additional wires. Consult a professional electrician for a technical analysis and advice. In some cases, swapping the existing box with a bigger one may allow you to install the additional wires.
At Least One of the Outlets is in the Kitchen or Bathroom
Lastly, you should not use an outlet for an expansion if it is a GFCI outlet in the bathroom or kitchen. Kitchen and bathroom GFCI outlets tend to have specific code requirements, and one of them is that they should be in their own separate circuits. This means that neither the new outlet nor the one you want to use for the expansion can be kitchen or bathroom GFCI outlets.
Therefore, if you are thinking of expanding your electrical outlet using an existing one, you need to evaluate it and ensure it will be safe. An electrical repair contractor should help with the evaluation to ensure maximum safety.