In federated learning (FL), a machine learning (ML) model is collectively trained by a large number of users, using their private data in their local devices. With top $r$ sparsification in FL, the users only upload the most significant $r$ fraction of updates, and the servers only send the most significant $r'$ fraction of parameters to the users in order to reduce the communication cost. However, the values and the indices of the sparse updates leak information about the users' private data. In this work, we consider an FL setting where $N$ non-colluding databases store the model to be trained, from which the users download and update sparse parameters privately, without revealing the values of the updates or their indices to the databases. We propose four schemes with different properties to perform this task while achieving the minimum communication costs, and show that the information theoretic privacy of both values and positions of the sparse updates can be guaranteed. This is achieved at a considerable storage cost, though. To alleviate this, we generalize the schemes in such a way that the storage cost is reduced at the expense of a certain amount of information leakage, using a model segmentation mechanism. In general, we provide the tradeoff between communication cost, storage cost and information leakage in private FL with top $r$ sparsification.