Bitcoin Core is technically the "official" reference client whereby all the protocol rules are defined on there. The project itself is open source and everyone can contribute to the project if they want and have the technical competency. I believe most "Bitcoin Developers" or the ones that are contributing actively are working in the field that's related to computing.
So the entire project is hosted on github[1] and there are repository maintainers to merge pull requests and someone in charge to merge to create new release candidates and it's stable release. Most of the changes comes from pull requests[2] whereby everyone discusses about the feasibility and the rationale behind the change in the code. They can comment to support the pull request or to state that they have reviewed the code. Afterwhich, the pull requests could potentially be merged by the repository maintainer.
You can also fork the client yourself to create your own version of Bitcoin, which makes it essentially an alt coin.
[1]
https://github.com/bitcoin/bitcoin[2]
https://github.com/bitcoin/bitcoin/pulls