High-dimensional data visualization is crucial in the big data era and these techniques such as t-SNE and UMAP have been widely used in science and engineering. Big data, however, is often distributed across multiple data centers and subject to security and privacy concerns, which leads to difficulties for the standard algorithms of t-SNE and UMAP. To tackle the challenge, this work proposes Fed-tSNE and Fed-UMAP, which provide high-dimensional data visualization under the framework of federated learning, without exchanging data across clients or sending data to the central server. The main idea of Fed-tSNE and Fed-UMAP is implicitly learning the distribution information of data in a manner of federated learning and then estimating the global distance matrix for t-SNE and UMAP. To further enhance the protection of data privacy, we propose Fed-tSNE+ and Fed-UMAP+. We also extend our idea to federated spectral clustering, yielding algorithms of clustering distributed data. In addition to these new algorithms, we offer theoretical guarantees of optimization convergence, distance and similarity estimation, and differential privacy. Experiments on multiple datasets demonstrate that, compared to the original algorithms, the accuracy drops of our federated algorithms are tiny.