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First, this assumes you already launched Skype normally from the Application icon, and logged into the first Skype account. One method is to open a Terminal (console) window, change directories to the Skype application, then simply run a second instance using the sudo command:
 
Step 1: Open a Terminal window
Step 2: cd /Applications/Skype.app/Contents/MacOS/
Step 3: sudo ./Skype
 
Now the second instance of Skype will launch, you will see trailing output in the console window.
 
Note: Do not close the window or it will force Skype to close. You may minimize the window if it's in the way.