I wouldn't worry about it unless the shiny part is significantly smaller than the original swept area of the pad.
The only remedy apart from replacing the disc would be to have it machined (skimmed) and you would need to change the pads But remember, with brakes, whatever you do to one side of the axle, you do to the other side to keep the brake balance.
So your options are, either replace the axle set of discs and pads, which is slightly more expensive but quicker, or have the discs skimmed and replace the pads, which would be cheaper but more down time whilst waiting for the machining of the discs.