Privatization. But your assumptions are wrong. Russians did have quite a bit of ownership before USSR fell and were in process of obtaining more when it fell.

 

If we talking about property in general, a lot of property was personal and owned in usual fashion. For example, individual could build (or buy) and own his own house/car/etc.

If we are talking about businesses (commercial enterprises), a lot of Soviet economy was owned collectively by workers (as co-ops), rather than state-owned. For example, kolkhoz - literally "collective enterprise" - specifically denotes this kind of non-state ownership and it existed since the beginning of USSR (but became commonplace during collectivization of 1930).

 

And, as I said, privatization was actually going on before USSR collapsed. Quite a lot of economic enterprises were already private (i.e. neither state- nor worker-owned, but 100% Capitalist-owned) - and those enterprises were private years before USSR fell.

 

In fact, collapse of USSR is often attributed (by Marxists; not Liberal economists, obviously) to this gradual privatization of state-run businesses (aka "Perestroika"): initially, both state control and oversight were relaxed, then more freedom was given to local management (aka "red directors") - uncontrolled distribution of funds and so on, but eventually even independent economic activity (up to and including banking, trading, private ownership of businesses - rather than collective, worker-owned) became legal and even encouraged - starting with the law "On Cooperation in USSR" (1988, May 26).

Privatization itself dates back to 1991, Jul 4 - when the law "On privatization of state and municipal enterprises in RSFSR" was enacted (which was, arguably, one of the main reasons why Soviet government had completely lost public support).

 

The actual implementation of the privatization was a mess. In theory, it was "buy state property from government": either directly, for money (this applied to smaller enterprises), or for privatization checks (commonly referred to as "vouchers" in Russia) population recieved from the government.

But, as I said, that's how it happened in theory. IRL corruption reigned. If enterprise was sold for money, it was usually fraction of it's value and often to foreign investor (as in DDR), while privatization via checks ended up with the aforementioned "red directors" (managers of state-run enterprises) seizing private ownership over most of state assets.

 

I hope this explains how Russians obtained ownership over state property.