The Halacha of "Vedaber Dabar" establishes various restrictions on the topics of which one may speak on Shabbat. Of course, there are certain kinds of speech such as Lashon Hara (negative talk about other people) that are forbidden at all times. But on Shabbat, special restrictions apply, including a prohibition against speaking about professional and commercial matters. These prohibitions are outlined and discussed by the Shulhan Aruch in Orah Haim, Siman 306.
One of the rules codified by the Shulhan Aruch is that "Hefseh Misva" issues involving a Misva may be spoken about on Shabbat. Even if the conversation relates to subjects that normally may not be spoken about on Shabbat, if one speaks for the purpose of a Misva, it is permissible. And thus the Shulhan Aruch rules (306:12) that one may announce a lost object, even if that object may not be handled on Shabbat. Common examples would be somebody who lost a pen, car key or cell phone in the synagogue, and so he asks that an announcement be made in the synagogue on Shabbat asking if any congregants have information about the lost object. Although these objects may not be used or handled on Shabbat, nevertheless, the announcement may be made on Shabbat, because (as the Mishna Berura explains) Hashabat Abeda returning a lost object to its owner constitutes a Misva. Spreading the word about a lost object thus serves the purpose of fulfilling a Misva, and it may therefore be done on Shabbat. This Halacha applies also if a Mukseh item was stolen and an announcement is made asking for information. And where Jews live together with non-Jews and there is a need to maintain peaceful relations between the two populations, it would be permissible to announce an object lost by a gentile, as well.
These Halachot are codified in Menuhat Ahaba (1:10, p. 218).
Summary: If somebody lost an object, an announcement may be made in the synagogue on Shabbat asking for information about the object, even if that object is forbidden for use on Shabbat, such as a car key, cell phone or pen.
