Mi Yodeya is actually a migration of an old Stack Exchange 1.0 site, https://judaism.stackexchange.com/, which you can still see in the Internet Archive: https://web.archive.org/web/20100424030814/https://judaism.stackexchange.com/
It can be seen in the first revision:
people for whom Jewish life and learning is a major part of their lives. (This proposal is for migration to SE 2.0 of http://mi.yodeya.com. Come see the great community and content we have already!)
As such, private beta was not required.
It was also said by the user who proposed the site migration in its meta, in response to "How come no one has the Beta badge on "mi yodeya":
The badge text is "Actively participated in the private beta." Mi Yodeya never had a private beta phase. There was no need, since the point of a community's private beta is " a 'probationary' basis to see if people will use it," and Mi Yodeya (actually Judaism.Stackexchange then) was founded on the pre-existing mi.yodeya community from the Stackexchange 1.0 platform. The initial user base was already there, as was a body of questions, a community culture, and provisional moderators. It would have been silly to close the site that was open to the public and open one that would be initially restricted to people who had jumped through certain hoops on Area51.