We propose novel data-augmented Markov Chain Monte Carlo strategies to enable exact Bayesian inference under the stochastic susceptible-infected-removed model and its variants. In the incidence data setting, where we are given only discretely observed counts of infection, significant challenges to inference arise due only a partially informative glimpse of the underlying continuous-time process. To account for the missing data while targeting the exact posterior of model parameters, we make use of latent variables that are jointly proposed from surrogates related to branching processes, carefully designed to closely resemble the SIR model. This allows several conditional sampling strategies that make classical MCMC ideas practical, surmounting the intractable observed data likelihood. The method extends to non-Markovian settings as well as tasks such as simultaneous change-point detection under time-varying transmission.