How do you remove stained silver nitrate from your skin?

Silver nitrate is non-poisonous. However, you still have to be careful because silver nitrate is irritant.

As a result, you have to remove it. However, soap is unable to remove it. To remove the stain, sodium chloride solution should be added to the stain. Then the stain can be removed by precipitation.

AgNO3 (aq) + NaCl (aq) → NaNO3 (aq) + AgCl (s)

Then silver chloride is being precipitated and can be removed readily.

The stain is silver metal , NOT silver nitrate!
Simply use IODINE to remove it.

The stain is not silver nitrate solution (silver nitrate solution is actually a colourless solution), but it is silver metal.

When siliver nitrate solution drops on your hand, you have to wash it immediately. After some time, silver nitrate solution decomposes in light to give silver metal, which is a deep brown stain.
2AgNO3(aq) → 2Ag(s) + 2NO2(aq/g) + O2(g)

The silver particles are very tiny. Therefore, it does not look silvery nor shiny. The silver stain will become paler in colour and will then disappear in one or two weeks (may be longer) due to metabolism of your skin.

Never try to remove the silver stain. When you use chemical (e.g. concentrated nitric acid) to remove the silver stain, your hand will be hurt.