Some first hand knowledge here. In some cases the suspected owner denies ownership ( few hundred pound piece found in woods, never claimed) Seems to match the maintenance reports that get filed months later of parts missing upon walk-round before next flight
A few cases where new owner demanded significant money for parts that fell before giving them back. Problem solved by immediate confiscation by the likes of the NTSB et.al. Not sure about new owner reusing the parts as long as they meet all air-regulations. One guy simply displayed the parts (fell off a truck0 in his front yard for all to see that he was within his rights.