- Under democratic communism, the government owns all housing
- Under democratic communism, each family owns their own housing
The key is that "ownership" is complex, a "bundle of rights" in modern (capitalist) legal theory. A more accurate description is that under democratic communism, the government has some rights from ownership's bundle, and occupants have other rights from that bundle.
What absolutely goes away is private absentee ownership of real property (land and land-fixed property such as buildings). To the extent that we have absentee ownership rights (which is necessary for property that outlives any specific individual's use), the government has those rights; to the extent that individuals do in fact occupy and use property, they have the ownership rights that accrue from occupation and use.
Thus, the government pays for the creation of new residential and commercial buildings (although they're actually built by private firms). The government at least pays for and probably directly maintains existing buildings.
Individuals have the alienable and revocable-for-cause right to occupy and use property. This right is alienable: the occupants can sell their right to occupy and use property to others. This right is also revocable for cause, on due process of law: an occupant has certain obligations and restrictions on the occupancy and use of property. The occupants cannot, for example, allow the property to deteriorate or be a hazard to others.
Although the right to occupy is alienable, it is owned only by physical possession. The monetary value of this right can be realized only by a transfer of possession. It cannot be owned or attached by anyone other than the occupant. (The right of occupancy can be attached by the government; in some circumstances, the government will probably provide credit to individuals to purchase the right of occupancy, and can, with due process, evict occupants for non-payment.)
Occupants pay "rent", more properly considered property taxes. Non-payment of rent (difficult, but not impossible) is grounds for coercive collection or eviction. (Eviction will occur only for relatively expensive housing when the occupants have had a decline in income.) The purpose of rent is to drain approximately as much money as the government injects to build and maintain property. The exact amounts collected in rent and spent in building and maintenance is a matter of government fiscal policy, dependent on immediate circumstances.
As best I can tell, the United States spends about \$1 trillion per year on all construction (residential, commercial/industrial, and government). Even trebling that number, it looks like total rent should average about 15 percent of disposable income, which is considerably less than the more-or-less modern standard of 33 percent of total income.
Every family is entitled to housing with rent of 15 percent of the minimum standard week's disposable income, within a reasonable distance of the working adults' jobs. There is residential rent control: a family occupying a dwelling cannot have its rent raised more than twice the inflation rate. There's no commercial/industrial rent control; raising rent is one way for local governments to force less-profitable firms to make way for more profitable firms. (The other way is to starve less profitable firms of circulating capital.)
For example, given a 24 hour minimum work week at 1 SVU per hour and a 12 SVU per week income tax, a two-parent family with two children would have an minimum disposable income of 36 SVU per week: 12 SVU per week per parent and 6 SVU per week per child. They would be entitled to housing at a rent of 15 percent of 36 SVU, or 5.4 SVU per week, or \$648 per month. Note that because rents are generally lower under democratic communism (since there are no landlords to extract economic rent), this corresponds roughly to a capitalist rent of about \$1500 per month, which is about the monthly rent of a small two- or three-bedroom home in the suburbs in my community.
Of course, people may spend more than the minimum on rent. If the parents of our example family above were to work 35 hours per week at 1 SVU per hour, then they would have a total disposable income of 58 SVU per week; if they spent 25% of their disposable income on rent (possibly including a loan for right of occupancy), they could spend \$1,740 per month for rent, almost three times the minimum.
All rent goes to the local government; all maintenance and construction is administered by the local government. The national government can add additional funding (from income taxes) for construction; they must do so if the demand for minimum housing exceeds the ability of the local government (which must run a balanced budget) to provide. If the people so choose, local governments can collect more in rent than they pay in construction and housing, with the extra money spent on other public goods (infrastructure, transportation, amenities, etc.).
There are still market forces operating on dwellings, but unlike capitalism, they are no economic rent-seeking incentives. If, for example, a lot of people want to move to a city (e.g. New York City or San Francisco), then they will be willing to offer existing residents a lot of money to purchase the right to occupy, providing an incentive for people to move out. They will also be willing to pay extra rent, giving the local government an incentive to build more housing. (Similarly for commercial/industrial rent.) Also, local governments can petition the national government for extra construction money; since (presumably) new residents of a city are being lured there by higher wages, they will pay more taxes, and it is in the interest of the national government to afford more housing to collect those taxes.
Unlike under capitalism, the asset value of all real property is exclusively in its ability to generate rent; since the government owns all real property, and cannot sell it, real property has no sale value. The government, therefore, has no incentive to leave real property unoccupied. (Indeed, it should probably be legal for anyone to occupy property unoccupied for more than a period of time (90 days?) and pay the minimum rent.) And, since rent in excess of maintenance and construction costs must be spent (local governments cannot save money), there is less of an incentive to raise rents to collect as much surplus value as possible.