Skype does not play well with others. It got its reputation for “always working” because it brutally takes over the computer sound services when it runs. That may seem to be evil behavior, but it lets Skype connect no matter how badly you messed up your sound. It also means no matter how carefully you set up your computer, Skype is going to change it. If you fight Skype, you could kill off your echo cancellation and one of the two sides will not be able to hear the other.
Skype conversations can be recorded with special purpose-built software packages. Pamela is the champ. The upper two licenses, Business and Professional will provide split WAV files for a Skype call suitable for post production/editing. You are warned about software that insists on using MP3 or jamming both voices into one file. Those can make post very difficult.
http://www.pamela.biz/en/products/
https://forum.audacityteam.org/t/help-recording-with-a-mic-laptop-and-headphones/35068/6
In general, self-recording is done by sending sound out to the sound card and then bringing it back in for recording. That’s why those techniques always use the Playback Volume control, and why it can sound so bad. The channel has all the soundcard playback and recording errors.
Some software creates a “fake” soundcard in memory and uses that instead.
http://manual.audacityteam.org/o/man/tutorial_recording_audio_playing_on_the_computer.html
Koz