The best duck hunting/habitat will be on private land, but there is an absolute ton of public opportunity to chase ducks in Missouri. A majority of the public, dedicated to waterfowl marshes are by draw only, which is a giant hassle, by the time the draw is over, its daylight, and your next back up spot might be an hour away. But, those marshes will hold a large percentage of the waterfowl in the state, and the bird per man averages are excellent.
There are many other smaller public marshes that do not hold draws, and they can be good as well. If you have a boat, the rivers and lakes can be outstanding, especially when we get a little rain, and our lakes get 3-4 feet above normal pool.
You can usually knock doors, and get permission to shoot geese in private fields, but you're going to have zero success accessing private marshes, for free. Duck clubs are big business in Missouri, with bad leases usually starting at over $1,000 per gun, and good leases well above that.
Best of luck, Missouri has tons of waterfowl opportunity, in just about any way you want to, or can hunt them.