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Private multi-winner voting is the task of revealing $k$-hot binary vectors satisfying a bounded differential privacy (DP) guarantee. This task has been understudied in machine learning literature despite its prevalence in many domains such as healthcare. We propose three new DP multi-winner mechanisms: Binary, $τ$, and Powerset voting. Binary voting operates independently per label through composition. $τ$ voting bounds votes optimally in their $\ell_2$ norm for tight data-independent guarantees. Powerset voting operates over the entire binary vector by viewing the possible outcomes as a power set. Our theoretical and empirical analysis shows that Binary voting can be a competitive mechanism on many tasks unless there are strong correlations between labels, in which case Powerset voting outperforms it. We use our mechanisms to enable privacy-preserving multi-label learning in the central setting by extending the canonical single-label technique: PATE. We find that our techniques outperform current state-of-the-art approaches on large, real-world healthcare data and standard multi-label benchmarks. We further enable multi-label confidential and private collaborative (CaPC) learning and show that model performance can be significantly improved in the multi-site setting.