2 360° overlayed videos

One simple way is to use a large, inside-out cube or sphere mesh that is concentric to the player's head. Don't make the mesh a child of the Camera - it usually works better to use a script to set the mesh's position equal to the Camera's without changing the rotation. A shader with alpha blending should work as usual there. Depending on how much code you want to write, you can either use multiple shader passes or separate meshes with an alpha blended shader for the different videos you want to display. For higher fidelity, you could use the multi-pass shader approach with Unity's Skybox component.