Recent efforts have aimed to improve AI machines in legal case matching by integrating legal domain knowledge. However, successful legal case matching requires the tacit knowledge of legal practitioners, which is difficult to verbalize and encode into machines. This emphasizes the crucial role of involving legal practitioners in high-stakes legal case matching. To address this, we propose a collaborative matching framework called Co-Matching, which encourages both the machine and the legal practitioner to participate in the matching process, integrating tacit knowledge. Unlike existing methods that rely solely on the machine, Co-Matching allows both the legal practitioner and the machine to determine key sentences and then combine them probabilistically. Co-Matching introduces a method called ProtoEM to estimate human decision uncertainty, facilitating the probabilistic combination. Experimental results demonstrate that Co-Matching consistently outperforms existing legal case matching methods, delivering significant performance improvements over human- and machine-based matching in isolation (on average, +5.51% and +8.71%, respectively). Further analysis shows that Co-Matching also ensures better human-machine collaboration effectiveness. Our study represents a pioneering effort in human-machine collaboration for the matching task, marking a milestone for future collaborative matching studies.