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
-p foo -no-remote.
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.