To send someone a song from your library privately, you must first head to the file library by clicking the top right "My Library" button in the menu while you are logged in.
From there, navigate to the song you want to wish to share with a friend in the "My Tracks" session, and look for the right-side "lock" button. Make sure this button is clicked so that the lock looks open.
From there scroll down to the bottom of the library page, and look towards the footer for a button titled "Copy Link." The button will be found to the right of the comment box.
From here, click the "copy link" button that appears from the popup, which will copy the track's URL for you to paste into a message or email to send to a friend: