Is it a shared system with other users?
By default, in Linux-based systems like Ubuntu your login can see any file in your home directory but nobody else on the system has access. Of course, the superuser has access to every file on the system. You also have the ability to encrypt your home folder, that can help prevent someone from removing your hard drive and mounting it on their own system (where they are a superuser) and getting access to your files.
Linux does not have per-app permissions, however. So any app you run has access to your Tribbler download directory and Tribbler has access to any file for the user you run it as. You can mitigate this by creating a special "tribbler" user which can protect you against rogue apps (or Tribbler) narfling your garthok.
Of course, all of this is about delays. There's a big difference between not wanting your spouse with a different login to find your porn cache and not wanting the DEA to realize you're a meth dealer.