If you’re aware of public and social housing then why are you asking how community ownership and management works?
In any case, yes, of course all rental housing should be publicly owned. Vienna’s Gemeindebauten and Singapore’s HDB, among others, have proven that pretty definitively.
I’m not certain that all housing should be public, though. Privately owned primary residences are probably fine, in the grand scheme of things. But rental housing for profit should obviously be abolished.
This is just a long-winded way to ask “how do we pay for it?” The answer is taxes. That’s always the answer.
Let’s call it 10 trillion total: 20m rental properties x 500k average home price. If we allocated half the annual military budget—400bn—to buying private rentals and making them social housing, it would take 25 years to get through the whole market.
The financial scale of the solution is not so large as to be insurmountable. The US government’s priorities simply lie elsewhere.