Short answer. The law requires you, as depositary, to advise the true owner once you discover the item was stolen and who that owner is. If the owner does not claim it within one month despite that notice, you are relieved of responsibility by returning it to the depositor.
What the law says
Nevertheless, should he discover that the thing has been stolen and who its true owner is, he must advise the latter of the deposit.
Civil Code, Article 1984 — No Duty to Prove Ownership; Stolen Things. Read the full provision →
You don't need to police the deposit up front
As a general rule, a depositary, the person holding an item for someone else, cannot demand that the depositor prove his ownership of the thing before accepting or continuing to hold it. Depositaries are not expected to investigate ownership before agreeing to safekeep something. That baseline changes only once you actually discover a problem, specifically that the item was stolen and you also know who its true owner is.
What discovering the theft obliges you to do
Once you have that knowledge, both that the thing was stolen and who its true owner is, the law requires you to advise the true owner of the deposit. This duty to notify only arises once both pieces of information are in hand; the statute does not require you to track down an unknown owner, only to inform one you have actually identified.
What happens if the owner does not respond
After you notify the true owner, the law gives them one month to claim the item. If the owner still has not claimed it within that period despite your notice, you are relieved of all responsibility simply by returning the thing to the person who originally deposited it with you. The notice you gave, followed by the owner's inaction within that window, is what clears you.
What the article does not say is what to do if the owner does come forward. It provides only for the case where the month passes in silence, and that is deliberate: once two people are both claiming the same thing, the depositary is no longer holding an uncontested deposit, and handing it to whichever of them sounds more convincing is a decision the article does not authorise. Where a claim is made, get advice before releasing the thing to anyone.
An alternative if you have doubts about the depositor
Separately, the law also allows you to return the item to the depositor if you have reasonable grounds to believe it was not lawfully acquired by that depositor, without necessarily waiting through the notice-and-one-month process. Either way, the practical first step once you suspect an item you are holding was stolen is figuring out, as concretely as you can, who the true owner actually is, since that knowledge is what triggers your duty to notify them.
Notice how differently the two routes are triggered. The duty to advise arises only on discovery of both facts, that the thing was stolen and who owns it, whereas the option to hand the thing back needs no more than reasonable grounds to believe the depositor did not acquire it lawfully. Suspicion lets you step out of the arrangement; only knowledge obliges you to reach out to the owner. Record what you learned and when, since the timing is what will be asked about.