People make products that work with Skype sound management to produce clear, high quality recordings of both sides. The Pamela upper two products will do that.
http://www.pamela.biz/en/products/
There are other programs, but be careful of getting locked into MP3 sound files (ratty editing) or programs that jam your conversation into one single track. Editing those can be very entertaining.
https://forum.audacityteam.org/t/help-recording-with-a-mic-laptop-and-headphones/35068/6
Koz