Manipulating droplets of biological fluids in an electrowetting on dielectric (EWOD)-based digital microfluidic platform is a significant challenge because of biofouling and surface contamination. This problem is often addressed by operating in an oil environment. We study an alternate configuration of sessile compound droplets having an aqueous core surrounded by a smaller oil shell. In contrast to the conventional EWOD platform, an open digital microfluidic platform enabled by the core-shell configuration will allow electrical, mechanical, or optical probes to get unrestricted access to the droplet, thus enabling highly flexible and dynamically reconfigurable lab-on-chip systems. Understanding droplet oscillations is essential as they are known to enhance mixing. To our knowledge, this is the first study of axisymmetric and nonaxisymmetric oscillations of compound droplets actuated using EWOD platforms. Mode shapes for both axisymmetric and nonaxisymmetric oscillations were studied and explained. Enhancement in the axisymmetric oscillation of the core by decreasing the shell volume was obtained experimentally and modeled theoretically. Smaller shell volumes reduce the damping losses, allowing the appearance of nonaxisymmetric modes over a larger range of operating parameters. The oscillation frequency regime for obtaining prominent nonaxisymmetric oscillations for different shell volumes was identified. Compound droplets provide a mechanism to reduce biofouling, sample contamination, and evaporation. We demonstrate axisymmetric and nonaxisymmetric oscillations of compound droplets with the biological core of red blood cells, providing crucial first steps for promoting applications such as rapid efficient assays, mixing of biological, fluids, and fluidic photonics on hysteresis-free surfaces.