You may see a message saying 'The user has chosen for this data to be private. No peaking!'.
To remove this message and allow recent activity to be displayed simple follow these steps.

Visit Bungie.net - link

In the top right corner click SIGN IN
Select your platform and sign in




Once your signed in, click your avatar to open the menu




Select Settings in the menu




Select PRIVACY.
You may need to scroll to the side if using a mobile device.




Make sure Show my Destiny game Activity Feed on Bungie.net is ticked
Then click Save



Your recent activity will now be shown on your profile.