This letter investigates single-target detection in an integrated sensing and communication (ISAC) system, implemented within a cell-free massive multiple-input multiple-output (MIMO) setup, based on a cloud radio access network (C-RAN) architecture. Unlike previous centralized approaches where sensing is processed in the central cloud, we propose a distributed approach where sensing partially occurs at the receive access points (APs). We consider two scenarios based on the knowledge available at receive APs: i) fully-informed, with complete access to transmitted signal information, and ii) partly-informed, with access only to transmitted signal statistics. We introduce a maximum a posteriori ratio test detector for both distributed sensing scenarios and assess the signaling load for sensing. The fully-informed scenario's performance aligns with the centralized approach in terms of target detection probability. However, the partly-informed scenario requires an additional 13 dBsm variance on the target's radar cross section (RCS) for a detection probability above 0.9. Distributed sensing significantly reduces signaling load, especially in the partly-informed scenario, achieving a 70% reduction under our system setup.