Miya: You should not think about who you are giving the lost back to, think about how it is the right thing to do and how it will the person who lost the object and how its a mitzvah you have to fulfill. You might not like the person who you are giving it to, even if you do not like that. That would be very difficult for me to do.
Micah: Just because its your enemy it doesn't mean you cannot help him with his animals. If you do not help it, the animal can suffer. It is like the burden of the animals is like a burden on your shoulders that will be relieved. The animal is not your enemy, the owner is. It is not right for it to be left to die.
From- "Jewish Lost and Found" by Rabbi Daniel Nevins, Torah with JTSA:
Finally, the Torah recognizes human nature—we are happy to help our friends and look out for their property, but the Torah warns us to do the same for our enemies. Indeed, the Rabbis say that one must give priority to returning the property of our enemies. In an audacious mode, the Rabbis say that the Torah’s entire purpose is to “subdue the evil inclination” (Bava Metzia 32b). I like to think that they were concerned not only with taming the petty instincts of a person who finds lost property, but also with restoring good relations between people who had been enemies. In this sense, the “lost possession” is friendship, and the commandment is to turn an enemy back into a brother.
From: Aish.com, Rabbi Shraga Simmons, "Returning Lost Objects"
Once you find an object, you’ll need to post signs in the area (or on Craigslist), and ask around for who may have lost such a thing. As an example, you would publicize: "Briefcase found on July 1st on Brookville Drive. To claim it, call 555-1234."
The key is to divulge enough information about the object so the owner will know it refers to him, but not too much information that someone could unscrupulously come and claim the object. Whoever calls to claim the briefcase would be required to give a “siman” – e.g. basic identifying marks like color, size, and perhaps some of the contents. In this way, we are certain that the object is properly returned.
Adam: The process of using sign or siman is a great way to help owners to find their things and get them back. And it is a good way to make sure a finder gets the things back to the right owner.
Jackson: In one of the versions of the movie Annie there was a rich guy that tried to find the rightful parents and he would pay a lot of money. But the next day thousands of people were waiting at the door. So, don't give away too much information or the people might fake their identity for that object.
The Talmud (Taanit 25a) tells the story of how chickens once strayed into the yard of Rabbi Chanina Ben Dosa. Rabbi Chanina thus became obligated to care for the chickens until their owner could be found. The chickens laid eggs which hatched into chicks -- and soon Rabbi Chanina's property was overrun with a whole flock of chickens! In order to consolidate, he traded all the chickens for a few goats.
Through careful observance of the mitzvah, Rabbi Chanina had multiplied the wealth of the original owner of the chickens. By the time the man came to stake his claim, he was the proud owner of an entire herd of goats.