Flexoelectricity is an electromechanical effect coupling polarization to strain gradients. It fundamentally differs from piezoelectricity because of its size-dependence and symmetry. Flexoelectricity is generally perceived as a small effect noticeable only at the nanoscale. Since ferroelectric ceramics have a particularly high flexoelectric coefficient, however, it may play a significant role as piezoelectric transducers shrink to the submicrometer scale. We examine this issue with a continuum model self-consistently treating piezo- and flexoelectricity. We show that in piezoelectric device configurations that induce strain gradients and at small but technologically relevant scales, the electromechanical coupling may be dominated by flexoelectricity. More importantly, depending on the device design flexoelectricity may enhance or reduce the effective piezoelectric effect. Focusing on bimorph configurations, we show that configurations that are equivalent at large scales exhibit dramatically different behavior for thicknesses below 100 nm for typical piezoelectric materials. Our results suggest flexoelectric-aware designs for small-scale piezoelectric bimorph transducers.