Most of current user profile related pages contain user-id in URL section, e.g. my edit profile page:
If you try to open it you will see a 404 stub (I hope you're not a SO stuff):
But it's not true for Preferences and Flair pages. For these pages it seems at a first sight that you can access another user's personal data, but I hope there are no security issues, just UI elements not linked to any data (you need to be logged in to see these pages). For instance here are Sonic's "preferences":
and Glorfindel's Flair:
Anyway I suggest hiding such weird pages and displaying the same 404 page as for other profile related tabs.
Bonus panel: for SO sites where theme support is enabled, changing the theme for not-yours page leads to displaying an empty popup from the top of the page: