Public vs private ownership of companies is a case by case basis. Many “capitalist” countries have many publicly owned companies. We used to have even more before Thatcher and Reagan. Now we have moved into a more public private partnership idea, which is a compromise.

Monopolies are able to extract monopoly rents through market power. This is one of the problems of rentier capitalism. That is why we have antitrust laws. We also need a system that prevents political rents from lobbyism for example by making it illegal for politicians to have stocks or to take campaign money from donors. We also have land rents from private land ownership. Singapore has a public land lease model, but a land value tax would achieve the exact same outcome.

In economics you talk about natural monopolies which is when initial investment costs are too high for competitors to exist or when physics or other constraints prevent competition (think of a railway line between two cities). There are many ways to argue that these types of companies should be publicly owned within a capitalist framework.

So yes, there is an in-between. And it depends exactly how much business is left to the government and how much is left to companies. This balance is defined by politics.