A socialist worker model of co-ops can provide competitive marketplaces as well.
Yes, co-ops are also a recognized type of organization in capitalist countries. They exist all over the world in and compete in the markets where they participate.
This seems like subjective definitions; cooperatives certainly work by issuing a segment of their ownership to each private individual who works for them.
This is up to them because they're operating in a capitalist model where organisations decide for themselves their ownership model and the allocation of their resources; contrasted with a central planning model.