If only a few people can own private property then you haven't abolished private property, you've just limited who can own it.
This was the main issue caused by the massive state control of Marxist Leninist countries like the Soviet Union -- by centralizing all control within one single political party you essentially just replace an economic elite with a political elite. Couple that with the democratic centralism you often find in these single party ML countries and suddenly you have something that's a lot less like the communal living the OP was referring to.
I'm not saying abolishing private property will be some magical fix all, but I also don't think using the soviet union as an example of why it won't work is a good example. You could have a society where usage of resources is governed by consensus rather than "lol stalin said so" but consensus is obviously a lot harder to do on a global scale than a small communal village