We develop bounds on the capacity of Poisson-repeat channels (PRCs) for which each input bit is independently repeated according to a Poisson distribution. The upper bounds are obtained by considering an auxiliary channel where the output lengths corresponding to input blocks of a given length are provided as side information at the receiver. Numerical results show that the resulting upper bounds are significantly tighter than the best known one for a large range of the PRC parameter $λ$ (specifically, for $λ\ge 0.35$). We also describe a way of obtaining capacity lower bounds using information rates of the auxiliary channel and the entropy rate of the provided side information.