One technique often used to deal with those "shiny spots" as well as many other skin blemishes is called frequency separation.

From The Ultimate Guide To The Frequency Separation Technique:

Frequency Separation technique is virtually a process of decomposing of the image data into spatial frequencies, so that we can edit image details in the different frequencies independently. There can be any number of frequencies in each image, and each frequency will contain certain information (based on the size of the details). Typically, we break down the information data in our images into high and low frequencies.

Like in music any audio can be represented in sine waves, we can also break up an image into low and high frequency sine waves. High frequencies in an image will contain information about fine details, such as skin pores, hair, fine lines, skin imperfections (acne, scars, fine lines, etc.).

Low frequencies are the image data that contains information about volume, tone and color transitions. In other words: shadows and light areas, colors and tones. If you look at only the low frequency information of an image, you might be able to recognize the image, but it will not hold any precise detail.

In a nutshell: frequency separation allows you to separate texture from color, particularly the texture and color of a model's skin, and work on each individually before combining them back together.

There are a plethora of online articles that discuss frequency separation and show how to do it with particular applications, particularly Photoshop CS. Many of the concepts can be translated to work with other tools, such as GIMP or other full orbed photo processing applications. Most of these tutorials are fairly involved and beyond the scope of distilling in an answer here. Be prepared to spend some time to learn how to do frequency separation. This isn't one of those "90 seconds to amazing images" photography tips!

http://fstoppers.com/the-ultimate-guide-to-the-frequency-separation-technique
https://www.youtube.com/watch?v=Qo6iBmYnqh8&list=PLZWkQI6iOhlBIlIosHjEH-b8ZViTyfRwB
http://www.retouchingebooks.com/retouching-skin-frequency-separation-technique/
http://www.creativebloq.com/photography/retouch-images-frequency-separation-5132640

Photoshop Elements: http://eliaslopez.net/blog/?p=245
GIMP: https://www.youtube.com/watch?v=wiWBYIr8-Kc
http://blog.patdavid.net/2011/12/getting-around-in-gimp-skin-retouching.html