I use Ubuntu, which should be pretty similar to Debian as far as this is concerned:

I literally just downloaded the zip file (or maybe a .deb?) and unzipped it in a directory in my desktop. All I had to do after that was make sure I didn't have both browsers open, and everything worked properly. They both used the same profile, and the tabs I opened in nightly were available in stable and vice versa.

I did nothing more than unzip the browser, and everything works perfectly.