As a counterpoint to the other answers:
Eating potato skin is generally not recommended, at least in Germany.
There are two main concerns:
- Solanine
- biocide treamtment (specifically anti-germination treatment)
Solanine
In Germany, most official sources caution against eating potato skin. The main concern is that the skin may contain elevated levels of solanine, a naturally occuring glycoalkaloid poison. For example, the German Federal Institute for Risk Assessment (Bundesinstitut für Risikobewertung) recommends:
Wenn Verbraucherinnen und Verbraucher Kartoffeln mit Schalen verzehren
möchten, sollten sie hierfür grundsätzlich nur unverletzte, frische
Kartoffeln verwenden. Insbesondere kleine Kinder sollten keine
ungeschälten Kartoffeln essen.
English:
If consumers want to eat potatoes with their skin, they should only
use undamaged, fresh potatoes. Small children in particular should not
eat unpeeled potatoes.
Speisekartoffeln sollten niedrige Gehalte an Glykoalkaloiden (Solanin) enthalten - Potatoes for human consumption should have low levels of solanine, BfR, 2018-04-24
The problem with solanine is that the quantity contained in potatoes depends on many factors (type of potato, growing conditions, age, storage). In addition to that, the level of solanine that is tolerable varies from person to person.
Therefore, the "safe" recommendation is to avoid solanine as far as possible, and that means discarding the skin, because solanine is more concentrated there. For example, this paper analyzed potatoes, and found that the skin contains easily 10 times more solanine than the interior part.
In practice, modern potatoes have less solanine than older varieties - in particular, the total solanine content (including the skin) is low enough that it is tolerable for most people. Therefore many people regularly consume potato skins without issues. However, there have been cases of serious (even fatal) poisoning in the past, some of which are listed in the document above. While often the potatoes had been stored improperly, at least some of these poisonings probably could have been mitigated by peeling the potatoes.
Anti-germination treatment
In storage, potatoes have a tendency to germinate (sprout), which spoils them. Therefore, the potato skin is routinely treated with biocides to prevent this. When eating the skin, these biocides will be ingested, which is a health concern.
Whether this is a problem in practice will depend on the exact treatment, of course. For example, in the European Union the substances approved for treating potatoes are believed to not be harmful even when eaten with the skin. In particular, for certified organic potatoes only Ethylen and essential oils are authorized. In that case the biocides should not pose a problem.
There is also a separate question for this point:
Risk of eating potato skins with anti-germination treatment
Summary
If:
- you are a healthy adult
- the potatoes are of a modern variety low in solanine
- they have been grown and stored properly
- they do not have green skin or a bitter taste
- they have not been treated with a harmful biocide
- you eat moderate quantities (<500g of potatoes per day)
the skin is very unlikely to harm you.
However, if any of these conditions are not met, there is a certain risk, which is hard to quantify. In particular, eating just the skins may be risky, since you could potentially eat much more skin than when eating it along with the rest of the potato.