What the hell, never knew about this:

  Consent to Access Your Files 
  BY UTILIZING THE SITE, CONTENT, FILES AND/OR SERVICES, 
  YOU CONSENT TO ALLOW DROPBOX TO ACCESS YOUR COMPUTER 
  TO ACCESS ANY FILES THAT ARE PLACED IN THE 'MY DROPBOX',
  'DROPBOX' FOLDERS, AND/OR ANY OTHER FOLDER WHICH YOU
  CHOOSE TO LINK TO DROPBOX.
  

Is this intended for the program to access the files and send them to the server? Or does this mean the bad thing, that nothing is really private?

Read it again, you will see that "Your computer" is then limited to "Files placed in" a certain location.

This is required for their service to work ...

I see the difference between Dropbox and Amazon terms, but I cant avoid asking myself If legally they are just the same.