Can Firefox be used with multiple Private Browsing sessions?

There is an add-on called MultiFox that does this.

OR, use the -no-remote command line switch to open another Firefox instance with its own profile if you want to run different Firefox versions simultaneously; you'll have to combine it with the -P switch specifying the alternate profile.

I have not actually tried doing this yet, but possibly try using Profile Manager to set up multiple profiles, and then the ProfileSwitcher addon would help with launching them? I'm pretty sure multiple profiles are not going to share cookies, anyway. I can't speak to whether it'll do anything weird with Private Browsing, but it sounds like you're really just using Private Browsing to simulate having multiple profiles open.

As of version 53 (or earlier) Firefox Test Pilot now has support for Containers, which is basically what you are asking for (a native version of MultiFox).

Unfortunately, as best I can tell it only works for normal browsing, so your private sessions are still all lumped together.