I think most of us can agree that taking the Bitcoin is morally wrong. However, what I'm most interested in here is what would you guys do. Leaving it there, knowing that someone probably hasn't left it there by choice, means that leaving it there you're probably allowing someone else to pick it up, and steal it. Although, by taking it (and not redeeming it) you're basically not allowing the person that have lost it to retrieve it.
You kind of answered your own questions there. I would take the piece of paper, but I would not take or move the coins. I know that my intentions aren't bad and I am OK taking it. I can't say the same thing about the next person that walks down the same path on the beach I did. I would then do all I can to find the wallet's owner, but even if I failed I wouldn't move the coins or consider them to be mine. I wouldn't give the paper wallet to the police. I have seen their competence when a friend of mine had his wallet stolen. Someone found it empty of all money and returned it to the nearest police station. The rest of their procedure is a joke I am not going to get into right now.
So, the moral stand point is if you know you are not going to redeem it, that's the only concrete information you have, everything else is chance. So, would taking the paper wallet, and then leaving a note of some kind for the original owner to contact you help?
On that same beach? I don't think it will help. Like you said yourself, you don't know where the wallet belongs. The tides could have brought it from kilometers away. The owner might not even know it was in the water in the first place. A robber might have broken into the real owner's house and picked it up together with everything else he found and not knowing what it was, he just tossed it into the sea.
What I yet don't understand is: What grants you ownership rights, if not the private key?
Forget about the law for a second. Someone generated that private key. That's the sole person who should have ownership. The private key allows you to move the coins but you are not the one who should have those keys. You took them/found them/stole them from someone/somewhere.