Where does "Save media" store pictures in Hangouts? In Hangouts (in KitKat) if you long click a picture in an MMS you can "Save media". Where is the picture saved to?
They go to your Download folder by default, typically at /sdcard/Download. Note that this may not necessarily be a physical SD card (e.g. on devices that don't have an SD slot it's emulated).

From a practical standpoint, if you need to access them on your device, just open the Gallery or Photos app. The picture will appear in either of those as long as you haven't added a .nomedia file to the Download folder or something.

Also, opening the picture in either Photos or the AOSP Gallery app (possibly others as well) will allow you to discern its physical location by opening the menu and selecting "Details".