I found this very interesting news article from May 2016, published by Kings College, London on the subject of "Hard water linked to risk of eczema in infants" . This was a study funded by the UK Food Standards Agency, Medical Research Council and the National Institute for Health Research. Here is a link to the full article, with references.
Atopic dermatitis (that is itchy, red, swollen, and cracked skin), pruritus (severe itching of the skin, as a symptom of various issues) and Eczema are some of the common skin reactions to hard water. Other not so well known effects on the skin include clogged pores and acne from what is left on the skin by hard water. In addition, more of your cleanser is needed to clean the skin and hair in areas with hard water because the abundant positively charged ions require a much heavier lather to dissolve.
A fifth of children in developed countries may be affected by Eczema. Factors affecting the surface of the skin and dry skin are thought to cause Eczema in early life, partly through genetics as well as being caused by environmental factors which may also contribute to the breakdown of the skin barrier, such factors do include water hardness and the amount of chlorine in household water (and, of course, swimming pools!).
There are various types of water softening systems. Perhaps the best work though an ion exchange system. All three minerals (calcium, magnesium and sodium) are positively charged ions called cations. The exchange takes place by passing water containing hardness minerals over man-made ion exchange resin beads in a suitable pressure vessel tank. During the ion exchange process relatively small amount of other strongly charged cations such as iron and manganese are also removed along with the calcium and magnesium.
Alternative less popular softening techniques include lime softening, chelating agents ("salt free" water softeners), distillation, and reverse osmosis.
For many issues, checking the hardness of your water and installing a water softening system is a positive step forward. As to which system may be best for you or in your area, professional advice should be sought although many healthcare professionals in the skin care field, suggest that an ion exchange system is the most helpful of all the systems available.