The most common way to receive this item is through using a Small White Sign Soapstone to engage in co-operation with other players. Successfully helping the host will reward the player with one of these items. They can also be found as treasure throughout Drangleic.
The Smooth and Silky Stone is used to trade for a random item with Dyna & Tillo who inhabit the bird nest in Things Betwixt. It can also be consumed to restore HP, although this is not the recommended use of the item.
Upon trading with Dyna & Tillo, which is done by choosing the Leave option in your inventory while standing in their nest, one item from a set list will spawn on the ground. The possible rewards can be found on the Dyna & Tillo page.
Smooth and Silky Stones may not be used to heal during PvP.