I think the money stays in your economy, increasing the wealth of your logging sectors, making them expand if private, or just giving you money back if not.

You are the country. The country has a Government, a Market, people, some other stuff; but they aren't you and they aren't each other. Your Government has a Budget, and your Market has Goods, and your people have wealth, and there's some other stuff. Money is not a Market Good, it's money, and if your Government or your People need Goods for something they can use money to add a Buy Order to your Market; adding the Buy Order lets the Government take the action that they needed the Good as an Input for.
The Market is a snapshot of current production (including some external trade if you allow it, your Market will track Trade as if the Goods are produced at the Trade Center), it is not a stockpile of materials. There are no stockpiles, except for money, although military Organization theoretically converts immediate Goods into a stat that then decays slowly over time when the input stops (doesn't seem to be completely working right now, but that's for a different post).

You still need to buy the wood from the building producing it but it will be much cheaper if you have surplus.