In this paper, we introduce a concept called "virtual co-embodiment", which enables a user to share their virtual avatar with another entity (e.g., another user, robot, or autonomous agent). We describe a proof-of-concept in which two users can be immersed from a first-person perspective in a virtual environment and can have complementary levels of control (total, partial, or none) over a shared avatar. In addition, we conducted an experiment to investigate the influence of users' level of control over the shared avatar and prior knowledge of their actions on the users' sense of agency and motor actions. The results showed that participants are good at estimating their real level of control but significantly overestimate their sense of agency when they can anticipate the motion of the avatar. Moreover, participants performed similar body motions regardless of their real control over the avatar. The results also revealed that the internal dimension of the locus of control, which is a personality trait, is negatively correlated with the user's perceived level of control. The combined results unfold a new range of applications in the fields of virtual-reality-based training and collaborative teleoperation, where users would be able to share their virtual body.