I believe Skype, like Zoom, has a recording service. They will record both sides of the conversation and send you the sound file. I believe at last check, Zoom will record both sides separately and send you the files.
That's the easy solution. As you're finding, doing it all yourself is an uphill climb.
Both Zoom and Skype take over your computer and there isn't a lot you can do about it without destroying the interview.
You can record you
on a stand-alone recorder and let the computer do the far side. Headphones are a really good idea. I use a Zoom (oddly enough) H1n recorder for local voice.
You can do it with two computers and a small mixer. That's what this is.
The right-hand computer is running Skype. Little does it know
that I sneaked the speaker and microphone feeds from the mixer rather than real life. The left-hand machine is running Audacity—recording either mixed or separate sound files. This is how the broadcasters do it when someone "calls in" to a show.
You can do this with three "people." One of the "people" is really a second computer who is logged into the interview but doesn't say anything. Its sole job is to record both sides.
I don't know of a good way for one computer to do everything. You're pushing against a multi-billion dollar company determined to do it their way.
If you run across anything, post back. It doesn't count if you find a one-off. There's a podcast done by a performer and his brother across the country. They use Audacity, just push record and go. Nobody else on earth can do that. He's looking at the rest of us like we're crazy.
There is one more two-machine solution. I put a special bi-directional microphone in my ear and plug it into a small recorder.
It will record me and anything I hold up to my ear (like a cellphone).
https://www.kozco.com/tech/audacity/cli ... dTest4.mp3