The Private Server is a form of MooMoo.io that is only available to the public via links that are often not disclosed to the majority of casual players. These are often times the exact same as their popular counterpart, and are used for complete solo adventuring or playing with a select group of friends or strangers. There were three private servers, two of which have been taken down, leaving one single Private Server to the public.
The first Private Server went by the name of Elastic Beanstalk, and was active until an undisclosed date. It was the same server used to take images of any weapons before the redesign of Update 0.31. It's strange namespace is unique compared to the other private servers.
The second Private Server went by the name of 188.8.131.52, and was active until well after the closing of Private Server ElasticBeanstalk.com. Information on this one is scarce due to it's extreme lack of use, although it was created after Update 0.31, and while it was being used, it was functionally identical to the main game. It also is at 1.4, despite being closed after 0.9. It's closure was just making it an infinite loading screen.
Although not classified specifically as a Private Server, the Dev Server shares the exact same properties of ElasticBeanstalk.com and 184.108.40.206, and so is featured on this page. It is currently the only open official private server, however, it is only at Update 1.4, so it will likely be deactivated in the future as well, or as a portal into the past of Moo Moo, like ElasticBeanstalk.com was before it's closure.