At the moment there isn't a way to turn on the "Private Session" mode automatically after log in.
However, you can also control how your listening habits are shared by going to Edit > Preferences (in Windows) or Spotify > Preferences (in Mac) and editing your 'Activity Sharing'. These preferences will be saved on your account until you decide to change them again.
@Meredith: You say "nothing will be shared" - Does this include the Spotify ticker at right hans site of the app? I mean what's the point of the "private session" button if you can just turn it all off in the preference settings?
I'm seriously considering making a fake FB profile to avoid the sharing stuff - I'm clicking the pricvate session EVERYTIME anyway.