Recommender systems have become very influential in our everyday decision making, e.g., helping us choose a movie from a content platform, or offering us suitable products on e-commerce websites. While most vendors who utilize recommender systems rely exclusively on training data consisting of past transactions that took place through them, the accuracy of recommendations can be improved if several vendors conjoin their datasets. Alas, such data sharing poses grave privacy concerns for both the vendors and the users. In this study we present secure multi-party protocols that enable several vendors to share their data, in a privacy-preserving manner, in order to allow more accurate Collaborative Filtering (CF). Shmueli and Tassa (RecSys 2017) introduced privacy-preserving CF protocols that rely on a mediator; namely, a third party that assists in performing the computations. They demonstrated the significant advantages of mediation in that context. We take here the mediation approach into the next level by using several independent mediators. Such distributed mediation maintains all of the advantages that were identified by Shmueli and Tassa, and offers additional ones, in comparison with the single-mediator protocols: stronger security and dramatically shorter runtimes. In addition, while all prior art assumed limited and unrealistic settings, in which each user can purchase any given item through only one vendor, we consider here a general and more realistic setting, which encompasses all previously considered settings, where users can choose between different competing vendors. We demonstrate the appealing performance of our protocols through extensive experimentation.