Modifiers have a few switches to control them:
The Camera icon enables or disables rendering using the modifier.
The Eye icon toggles visibility on the 3D viewport.
The Dotted Square (or Box) toggles visibility in Edit mode.
The Triangle, when enabled, will show the final modified geometry in Edit Mode and can be edited directly. (This last option can only be used when visibility in Edit mode has been enabled)
To learn about all of the other options for modifiers please read:
Note that the mirror modifier will affect all of the mesh when enabled. The elements that you don't need to mirror will need to be a separate mesh. There is no way to mark elements of the same mesh not to be affected by the modifier.
You might want to separate the asymmetrical features from the original mesh, then
Apply the modifier to make the symmetric mesh a permanent one, and re-join the two meshes back together.