I have plane object with texture material and bothSides set to true. Once I add light and shadow to scene I notice that on the back side of the plane you can see light and shadow which is dropped on the front side of the plane. How can I make the plane not so transparent for light? I tried to make two planes with little distance between but new bugs appear.

You will have to make two separate planes because the normals of the plane can only point in one direction.

What bugs are you seeing when you have two planes “back to back”?