This question inquires about private ownership vs. state ownership. However, there is a 3rd category that may or may not be considered as private ownership: Worker cooperatives.
And a fourth, and a fifth, and so on. Who owns property is not a distinguishing feature. States run market capital all the time. Private companies run sheltered workshops. Workers coops maximise value and circulate it in the expanded form. Public, private or cooperative ownership is not a predictive indicator of capitalism, socialism, firm structure, market conduct or any other differentiator.
In the model of private or public ownership, where would worker cooperatives fit in?
They're good evidence the model is faulty.
Is this ownership model closer to socialism, closer to capitalism, or is it its own category entirely?
Neither. Again it is a faulty model. Mondragon produces capital in an expanded form. The Rochdale Coops did too, the divvy was a distribution of capital. What is important about workers coops is the increase in power for workers within capitalism, the possibility of different firm structures by a potential to offer more diverse management values than maximisation of shareholder profit, and the presence of non-capital expanding economic conducts. But these can happen in other ways, local entrepreneurs often devote personal income to local interests, because of the latent threat of local consumers and workers against them.
What political movements promote this form of ownership?
Social democracy does in some areas. Anarchism, as in class struggle anarchism, does also. Revolutionary social democracy, your classic Leninisms tend to avoid the issue. As do labourisms (Australian Labor Party, Labour in the UK, etc).
For the bonus, 22% of Swedish housing stock is provided cooperatively. http://www.housinginternational.coop/co-ops/sweden