Private Registries for ROS Packages?

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The typical answer to this would be to make use of ros_buildfarm and related packages to setup your own private instance of a ROS buildfarm (similar to build.ros.org). Nothing would have to be public in that case (although setting up deployments from private GH/BitBucket/etc repositories is something that has a few rough edges (see the issue tracker)).

There have been a number of presentations about the ROS buildfarm at ROSCon, The ROS build farm - what it can do for me (slides) is one of those. It's from ROSCon'16, but probably still provides a good overview.

There are various alternatives to the buildfarm, ranging from manually catkin build-ing a workspace and tarring that, all the way to using deployment platforms like snap and mender to "bundles" (slides), buildbot-ros, multi-stage Docker setups (Hermetic Robot Deployment Using Multi-Stage Dockers (slides)) and many more.

Which one of these approaches works best for you is something I don't believe we can answer here, but if you have specific questions about any of them, I'd suggest you post additional questions either here or on the trackers of the respective projects.