Suppose, for example, that Person A drops a valuable ring and a sticky note that happens to say, “Free” on their driveway. Person B walks by, sees the ring, and takes it without notifying Person A.
What can Person A do in this scenario or other similar scenarios where Person B, either reasonably or not, takes something that they believe Person A is giving away?
Can a store sell merchandise I've left in the store? is related, and explains that the answer depends on the concept of lost, mislaid and abandoned property. Wikipedia summarises the general common law rule in these terms:
A finder of property acquires no rights in mislaid property, is entitled to possession of lost property against everyone except the true owner, and is entitled to keep abandoned property.
Person A's property meets Wikipedia's definition of lost property:
Property is generally deemed to have been lost if it is found in a place where the true owner likely did not intend to set it down, and where it is not likely to be found by the true owner.
Person A's property may also meet the definition of abandoned:
Property is generally deemed to have been abandoned if it is found in a place where the true owner likely intended to leave it, but is in such a condition that it is apparent that he or she has no intention of returning to claim it.
In this case, the objective circumstances strongly suggest that the ring is abandoned. This means that Person B owns the ring and Person A can't get it back.
However, it is stipulated that A did not intend to abandon the ring. This suggests it was merely lost. If a court finds that the ring is lost rather than abandoned, then A, as the "true owner," can sue B to recover it.
My reading of this question is OP trying to figure out if there is a cloud on B's title to the property or not. That is, is there anything A can do legally to reclaim any title to the property?
The primary answer to that is that all A can do legally is try to show that the property does not meet the definition of 'abandoned' in sjy's answer. For example, A could deny putting the property at the end of their driveway with a sticky note saying 'free.' Even if B had a photograph of the ring in that location with that note, A could try to claim that B had previously taken the ring in another time and place, then created the note and placed it at the end of the driveway themselves just for the picture.
It would seem quite odd for A to both actually do what is described, and then pursue a formal legal action to try to reclaim the item. A has no apparent motive to do this, but if B obtained the property by another means (e.g. stealing it from A's pocket/house/hand), B would have a clear motive for fabricating the note and photograph, which would be relatively easy to do.
A's attempt at a legal action for recovery would likely be in the form of a civil suit, where the standard is typically "a preponderance of the evidence," or "more likely than not." For A to prevail, A would only have to convince a judge/jury that their denial of putting the 'free' note on the ring and leaving it at the end of their driveway is more likely than not to be true. To discredit that, B would in practice likely have to at least offer a plausible explanation of why A would do such a thing and later deny it. The case would come down to relative credibility of the testimony and other evidence.
A could also report this as a theft to the police. A conviction for the theft would require A (with the police and prosecutor) to prove beyond a reasonable doubt that A didn't do what B describes. That might not be very hard in a situation like this, especially if there is other evidence of some form of theft.
In practice, especially if A and B know each other and anticipate future interactions, the most practical outcome would be that even if A admits to putting the 'free' note on something and leaving it at the end of their driveway, if B still has possession of the item when A requests it back, B ethically should return it. If A makes a habit of doing this which is bothersome to B, B can stop picking up the items A leaves out.
