You (or anybody else) should be able to run a copy of service (including all of its data) under your own control.
Examples: Wikipedia has open source engine and DB dumps available, thus allowing anyone to fork the project.
Counter-examples: Juick has proprietary engine and DB dumps are out of question.
If service doesn't need to understand your data, you should be able to encrypt it and don't trust service with decryption key.
Examples: Amazon S3, GMail both can be perfectly private if you store encrypted data only.
If you see a bug, you should be able to fix it and, optionally, submit a patch, which then should be accepted.
Examples: Sauerbraten has open-source server and lots of mods are available.
Counter-examples: Counter-Strike has proprietary server which behaviour can only be slightly modified via scripts.