You can use symmetry to mirror sculpting operation from on side to the another.
The option to enable/disable the symmetry can also be found on the left quick panel ("Sym").
Because of technical reasons, some brushes will have imperfect symmetry behavior.
In that case you can use the Mirror button to apply back the symmetry from one side to the other.
This operation only succeeds if the algorithm detect that the mesh is symmetrical. In case the detection failed, you will be asked if you want to force the mirroring.
The main difference is that forcing the mirroring will change the topology, thus you might loose the Mutiresolution.
# World vs Local
By default the symmetry is in World space, meaning that the symmetry plane is fixed, even if you translate or rotate the plane.
You can see in the video below the usage of World/Local space and the behavior of the Mirror feature.
# Editing the symmetry plane
You should probably never use these options but you can reset the symmetry plane origin and orientation, in case you moved them.
Note that the behavior of these options will change depending on what space (Local/World) you are in. So if it doesn't work as you would expect, make sure to check if you are in the correct space.