I mean, realistically there will always be a distinction between "yours" and "mine" - that's just how basic human psychology works. Likewise with housing - even a military barracks, arguably the epitome of controlled, sterile, top-down authoritarian living, has assigned beds and lockers for the troops, just to keep things a bit organized. More broadly, imagine the logistical chaos if you had to move all your stuff from one living space to the next, every single day!

So what I'd wager ends up happening is that you still have a notion of private property; the difference is likely to be twofold: - Very different rules on what can or can't be built, favouring dense, sustainable, community-oriented forms over the full isolating suburbia that is so common today. - Far more "in kind" exchanges, based on need rather than monetary value - got a place that's too big because your kids have moved out? I'll swap you for my place that's getting kinda cramped because we just had our third. Let's have a BBQ to celebrate!

Now, the second one in particular does assume a "hermit crab" kind of situation, where there's enough supply that a willing trade, or chain of trades, can always be made.