Unclaimed paintings are sold at auction with an automatically decreasing price mechanism. As the price decreases over time, the first player to buy a painting gets it at the current price. Paintings are held by a player until they are stolen by another player. Paintings are displayed in the gallery area on the left side of the screen (though they are stored in the vault). If the current owner of a painting dies, the painting put up for auction again with a starting price of $100,000. There is no way to sell a painting.
All paintings were made before the game launched by Jason's close friends, relatives, and indie game design colleagues. 
The user "iceman" maintains a thread containing snapshots of where paintings were located at certain times, alongside a crowdsourced list which will usually be more up to date. This is not guaranteed to be 100% accurate, as paintings are changing hands all of the time, but is a good place to look if you intend on stealing a certain piece of art.