Eczema is a condition caused by an over-reaction of the immune system, not by a bacteria or a virus that can be passed between people. People with eczema are generally born with it ~ and start showing signs of red, itchy and flaky skin before they start school. Irritants and allergens in the environment can make the condition worse ~ as can stress and scratching.
Once the skin is broken, there is a risk of infection, which can cause more misery as the skin develops open sores that bleed and weep. Care should be taken to treat any infections rapidly, and to cover any open skin sores until they can heal. Eczema may not be catching but fungal and bacterial infectious can be.