Update 2016-08-06: Firefox now provides an about:profiles page which can be used for creating new profiles. This supersedes Step 1 below, but I haven’t yet tested how it works.

Thanks to @vWil's advice I was able to finally make it work. Here is my step-by-step guide. Note: I'm performing this on Windows 8.1. I'm not sure if the process is different on OS X.

Step 1: Create a second profile

Note: Firefox must be closed while performing this step.

Press Win + R to open Windows’s “Run” dialog, then type "firefox.exe -p" into the input field and press Enter .

This should open a "Choose User Profile" dialog:

Click on the "Create Profile" button and follow the steps. Use a simple name for this profile. I've chosen "nite":

Step 2: Install Firefox Nightly

The download page is here.

Step 3: Set command line flags

Locate the launch icon for Firefox Nightly, right click on it and go to Properties:

In the Properties dialog, add -p nite -no-remote to the Target field:

If you've set a different name for the second profile, use that name instead of nite , e.g. -p foo -no-remote .

That's it

Open Firefox, then open Firefox Nightly using this custom launch icon. Both should be able to run at the same time:

Please let me know via comments if this process didn't work for you.