This should give you half the answer you're looking for - you'll hear the game, your own voice, and their voice. Your friend will hear you, the game, and themselves. It's not the "perfect" answer you're looking for, but I personally do this and it works quite fine.
Simply go to your Sound Settings and configure your mic as shown:
![enter image description here][1]
By enabling "listen to this device", you're broadcasting your mic directly to your Stereo Mix. Likewise, your game and your friend's voice are also being broadcasted to the Stereo Mix. Open the Windows mixer and adjust the levels to ensure that all three sources can be heard, then set the Skype input to Stereo Mix to send everything to your friend.
If your soundcard has two independent headphone outputs, then it might be possible for you to not hear yourself (and your friend can't hear themselves either). Set the Skype audio output for the first headphone output, so your friend's voice will feed into that jack. Then, set the game's sound and your voice out the second jack, and route that signal (using Stereo Mix) to the Skype input. Use a 3.5mm y-connection to mix both outputs into your headphones, and everything should be all good. :D [1]: https://i.sstatic.net/ZgXdr.png
