IANAL, but just a web searcher. It appears that legally, in the US, public-facing private spaces are in a weird place. A lot depends on the specifics of the space. The closes analogy I can think of is shopping malls, which are privately owned but used as if they were public spaces.

In general,

"Private spaces used by the public, like shopping malls or private campuses, are typically non-public forums. The government cannot compel access to private spaces, but private operators may allow or restrict expression under private law and policy, balancing property rights with constitutional protections in some cases."

However, there have been cases where constitutional free-speech protections were extended into malls. This can happen if the property takes part in federal programs or gets federal subsidies or tax breaks.

https://bridgelegal.org/what-legally-defines-public-space/

If we want a digital public square we should fund one with public tax dollars.