I'm going to regret getting into this. Not least because I know full-well I don't have an ultimate answer.
But I do not get the strict division you make between 'private' and 'the state'. How do you justify that dichotomy? What makes a state a state, rather than just a very big private corporation that happens to own all the land and/or all the 'intellectual property'?
In Russia the two seem to have merged. Seems to me that post-Soviet history demonstrates the truth in the critcisms of anarcho-capitalism (ironically, a criticism that Ayn Rand herself, more a minarchist-capitalist, made of those like David Friedman who would outflank her on the right). That you end up with a private monolith that owns the police and justice system, and which becomes indistinguishable from a state.
And, related to that, how do you determine who owns what in the first place? Especially when it comes to land and ideas.
Its fine. You should realize by now that I'm not the type to go after people for having a conversation.
So, what makes the state different than private firms is how they gained their property. Private firms gain their wealth/capital through free exchange with all willing groups. The state gains their wealth/capital through collective action where consent is not required. Those are the fully pure forms though, and you are not likely to find that any exchange is either or.
As for what is ownership and how can it happen, that is a massive question that I wont be able to answer fully tonight. Property rights is super fun to get into though, so lets do it tomorrow.