Well, ... forget for the moment who has "jurisdiction" over any waterway (city, county, state, fed, or private). And concentrate on the other part of your question which is about "digging" and "ownership of what's found". If you want to worry about those two things, then it becomes irrelevant which type entity has ownership/jurisdiction. Because the minute you start asking about "digging", I can bet that somehow ... someway .... there's going to be some pencil pusher that can say "no" to that. Even if the word "dig" is not listed as a prohibition, they could lump it into another category. Like "deface" or "alter", etc... But .... seriously now .... does any one really care (until you ask) ?
And as for the ownership of items, I can save you time there too: For items worth over a given dollar amount, in any state found on ANY form of public land, they must be turned in to the police. The value is usually something like "$100 or more..." Those are "lost & found" laws born out of wandering cattle laws of the 1800s. But the last time I looked at any of the "finds" forums, there's no shortage of md'rs proudly boasting their latest trophies (rings, coins, etc...). Do you think they're trotting down to the police to turn each one in ? And even if not for "lost & found" laws, there's always something, in any city, county, state, or federal laws, that forbid "harvesting", "collecting", "taking", "removing", etc.... I mean, so that obviously no one thinks they can help themselves to the picnic benches, or start harvesting sod out of the park for commercial sale, etc... Can such verbage be applied to individual coins and relics? Sure! Just ask enough bored lawyers or state clerks, and someone will morph those to apply to your "pressing question". But .... seriously now.
If it were me, I'd just detect said-waterway. Afterall, aren't you looking for the ring your wife lost there yesterday ?
