Re: Current River
Much of the Current from Montauk to about halfway between Van Buren and Doniphan is part of the Ozark National Scenic Riverways, and all the land within the Riverways is owned by the federal government (National Park Service). So anywhere you can get to it, you can wade and walk the banks to your heart's content. It is not, however, wader friendly below Akers Ferry. The trout sections, from Montauk down to Akers, are wadeable most of the time. And by the way, there are a few smallmouth below Ashley Creek a few miles downstream from Baptist Camp Access, and they get a little more common below Cedargrove. But really, they aren't numerous enough to furnish consistent fishing until you get to Round Spring. The smallmouth sections of the Current are too big to be easily wadeable.
The river downstream from Cedargrove gets a LOT of canoe and kayak use during the warm weather months, and below Round Spring, you'll encounter a lot of jetboats on warm weather weekends. On the lower river below Van Buren, the river can be crazy with high speed jetboats. It's big water down there.
If it's smallmouth you want, however, there are other choices nearby. The Jacks Fork is also a part of the Riverways, and is a nice smallmouth stream, with much of it wadeable in the low summer water levels. There are also interesting, if remote, smallmouth possibilities in some of the smaller Current River tributaries, but you'll have to research it yourself, since I'm not too keen on publicizing such fragile waters.
As for Missouri stream access laws, basically, if it's floatable by canoe you can legally wade it as well as long as you get on it at a public access. If it's too small to float, you cannot legally wade it if a landowner doesn't want you to. Many wadeable streams are still "open", simply because the landowners haven't yet decided to keep people off them. The landowner owns the banks and stream bottom, but on floatable streams, court cases have affirmed a public right to use the stream for floating, wading, and even camping on gravel bars.