.... Hypothetically. I'm not actually going to, but I've been working on a thing, and it'd involve similar situations.
Hey there tinted-snow, you're talking about public land, yet where a private person has made a "claim" for mineral hobbyist purposes (nuggets, panning, sluicing, etc...). Right ? I'm not into nuggets in-their-natural form, on public land. Where a private citizen has made a "claim". So I can't answer to the exact legal question.
But assuming, for the moment, that the person holding the claim has A) rights to whatever old jar or tin filled with coins that might exist there, and B) Is even there, in the first place, to make a fuss about it , then : I've been on many many many farmers and homeowner's lands with permission. Which I presume to be similar to your question of "claims" and "mineral rights", if we assume they own any pebble, or can, or jar, etc... that's there.
And with that said, in my 40+ yrs. of this, I have NEVER had an issue of splits, or ownership (eg.: something "ripped out of my hands") occur. In each and every case, it's generally a friendly smile, a handshake, etc.... And on occasion, if I show them a cool old coin or token, they tend to say "take it". Of course, I always offer the things to them (and have even made extensive shadow box displays, etc... for them). And I'll usually hold out a single item (token, button, etc...) and say "This is one that I don't have in my collection, can I keep this one?". And have never had anyone say "absolutely not".
Granted, I haven't had a cache be the issue. Rather: Just fumble fingers coin-shooter stuff. And granted, Even IF someone grabbed a cache or a jar from me and said "you can't have that" and "it's mine", I would STILL not have regretted the hunt. As long as I have some pix for my own trophy hall, and had fun, I'm fine with that. And, granted, I still have "clothes on my back", "food in my stomach" and a "roof over my head". So why would I get upset with them, get legal on them, etc... ?
And to whatever extent someone could nip such a situation in the start, by making extensive "search and salvage" legal agreements ahead of time : To me, that seems like the FASTEST way to get a "no".
When I go to search someone's cow pasture (where a stage stop existed) or their yard, etc..., I'm fully aware that I might have to turn everything over. But that has never happened.
So your question is one of those type of questions where, admittedly, if you turned over heaven and earth, it has never-ending ramifications of doom and gloom (ie.: why bother?). Yet the human reality of md'ing, finding stuff, etc.... is most often far different. I'm sure there's the fluke random "horrible stories" that will circulate. Of someone who was "robbed of his findings" and thus "went hungry and naked". But .... if we concentrated on those fluke stories, we'd never leave our front doors in the morning.