Existing abstractive summarization models lack explicit control mechanisms that would allow users to influence the stylistic features of the model outputs. This results in generating generic summaries that do not cater to the users needs or preferences. To address this issue we introduce HydraSum, a new summarization architecture that extends the single decoder framework of current models, e.g. BART, to a mixture-of-experts version consisting of multiple decoders. Our proposed model encourages each expert, i.e. decoder, to learn and generate stylistically-distinct summaries along dimensions such as abstractiveness, length, specificity, and others. At each time step, HydraSum employs a gating mechanism that decides the contribution of each individual decoder to the next token's output probability distribution. Through experiments on three summarization datasets (CNN, Newsroom, XSum), we demonstrate that this gating mechanism automatically learns to assign contrasting summary styles to different HydraSum decoders under the standard training objective without the need for additional supervision. We further show that a guided version of the training process can explicitly govern which summary style is partitioned between decoders, e.g. high abstractiveness vs. low abstractiveness or high specificity vs. low specificity, and also increase the stylistic-difference between individual decoders. Finally, our experiments demonstrate that our decoder framework is highly flexible: during inference, we can sample from individual decoders or mixtures of different subsets of the decoders to yield a diverse set of summaries and enforce single- and multi-style control over summary generation.