סִימָן קפז – הִלְכוֹת אֲבֵדָה וּמְצִיאָה
186: Laws of Lost and Found Items
Someone who sees something lost by a Jew, he is obligated to busy himself with it to return it to its owner, as it says, “[Do not see your brother’s ox or his sheep driven away, and you hide yourself from them;] you must bring them back [to your brother].” (Devarim 22:1) Similarly, any of his friend’s money that a person can save from getting lost, he is obligated to save, and it is within the category of “returning lost items”.
Even though by the strict law, in a place where most of [the people] are non-Jews, even if a Jew put on his item a sign [the finder] isn’t obligated to return it — because the usual situation would be that [the one who lost it] gave up on it [being returned] — in any case it is good and right to go within the line of the law to return to a Jew who put a sign on the item. And we compel someone to do so. If the finder is a poor person, and the one who lost it is wealthy, he isn’t requires to go beyond the letter of the law.
In a place where there is a civil aw requiring the return of lost items, he is obligated in any situation to return it [– Jew or non-Jew, sign or no sign].
As you may have noticed from previous comments, e.g. about charging interest (65:1), my rule of thumb is that if the action is not obligatory/prohibited where the other party is a Jew then the issue isn’t one of monetary fairness, but of brotherhood.
I therefore would conclude from here that — barring local law — “finder’s keepers losers weepers” is a valid fiscal principle. However, when local law is that a person retains ownership of a lost item, then a Jew would be obligated to return the item for two reasons:
1: Local custom can define fiscal norms, and therefore the law means the non-Jew still owns the item. To use it would be theft.
2: There is a halachic obligation to be a law abiding citizen (assuming the law neither calls for a violation of halakhah nor is anti-semitic in intent).