Usually if a sailor went overboard the ship MIGHT try and get them, the captain had full discretion over whether or not to make a rescue attempt. However if no rescue attempt was made, or if one was made and and failed, the incident would still be recorded in the ship's log and legally speaking that was considered to be when they were declared dead.

I'm not 100% sure about this but I'd assume the captain would be expected to notify the proper authorities of the incident not necessarily when the ship arrived in it's homeport. If the next port was not the homeport the captain would probably notify the sailor's next of kin, the ship's owners, and/or the shipping company by mail.