The people own the means of production.
how do you instantiate this ownership
Here's a little sneak peak at coming attractions:
The proletarians have nothing to lose but their chains. They have a world to win.
Working Men of All Countries, Unite!
Now, I guess for certain very peculiarly small notions of "state" you could say that a consortium of workers joined together could represent a "state", but that is stretching it. The purpose of all this is really that the people who are doing the work are getting crapped on by those who in many cases did nothing more noble than crawl out of the "right" vagina; Marx wanted to see the people who actually do things receive a better part of the pie and a better chance at their own destiny.
Property and means of production are not the same.
One may be a subset of the other
Only in very specific cases. Very few people make their living by pressing a start button on an automaton in the morning and waiting for saleable widgets to pop out by the end of the day.
was Marx for or against private property?
Marx was not opposed to private property. What would be the purpose of making anything if nobody was allowed to own anything? He did not aspire for the workers to rise up only to themselves shut down the factory and go become peasant-farmers.
Marx's opposition in regards to private property lied in its runaway accumulation and how it aided in the oppression of workers, but he did not oppose private property on its own.
To ask such a binary question is absurd, really. If everyone must be either opposed to it or in support of it, then anyone in support of it must be in support of all forms of it. Would a conservative support private ownership of nuclear weapons? Would a conservative support the private ownership of an entire state or nation? Would a conservative support private ownership of other people as private property?