By "open source", which is mainly a term used in software development, I believe you're referring to open standards, which are systems (like a camera mount) that aren't specific to a certain corporation or company, but are allowed to be legally used by multiple corporations within an industry.
Camera mounts don't have to be open standards for third-parties (like Sigma, for example) to make lenses for that mount, as long as they aren't patented, which from what I read, it appears that all modern camera mounts aren't. Companies can license their camera mounts to third-party manufacturers, or third-parties can reverse engineer lenses.
From what I read, this is what I gathered on the major camera mounts:
- The Nikon F-mount and Canon's EF mount are both proprietary, though third-parties are allowed to reverse engineer their lenses for compatibility legally. Canon discourages the use of third-party lenses and doesn't appear to have ever licensed their mount to any third-party. Nikon apparently has licensed their lens technology to Tamron and Tokina; Nikon a couple years ago sued Sigma for patent infringement, though it wasn't over lens mounts.
- The Pentax K-Mount was originally an open standard (see here), and has been used on many cameras from different companies. Today there continues to be no restriction on third-parties manufacturing lenses for the K-mount, but I believe Pentax is the only company still making cameras for the mount
- The Micro Four Thirds system, which today is primarily used by Olympus and Panasonic, is an open standard, and is allowed to be used by any manufacturer
Also see this similar question that may help: Why don't Canon and Nikon lock out third-party lens makers?