Foldit offers two types of screenshots. One type is saved locally on your computer, and the other type is stored on the Foldit server and shared with others, as described in posting results.
In the main menu (or the "menu" menu), there's a Save Screenshot button. The shortcut key control + h performs the same function.
Screenshots saved this way are stored locally on your computer. A popup message in Foldit tells you the file name and location.
The location is controlled by the "gui/image_dir" keyword in options.txt, for example:
"gui/image_dir" : "C:\\Users\\*username*\\Documents\\Foldit"
On Windows this setting would store screenshots in C:\Users\*username*\Documents\Foldit, the equivalent of My Documents\Foldit for user *username*.
There's also a camera icon in each of Foldit's in-game chat areas. Clicking on this icon uploads a screenshot to the Foldit server, and posts a link to the screenshot in the corresponding chat area. This is how players upload puzzle results for others to see.
For example, if you click the camera icon under "Chat - Global", the link to the screenshot is posted in global chat, and anyone can see it. Screenshots from "Chat - Veteran" can be seen by anyone in veteran, and screenshots in "Chat - Group" can only be seen by the members of your group who are currently connected to chat.
The links created by the chat cameras are visible only to those who are connected to the chat area when the links are posted. The links may be shared with anyone, however. For example, the link to a screenshot posted in group chat can be shared with anyone, group member or not, Foldit player or not.
Chat screenshots do automatically blank out the contents of other chat windows, a helpful featured for preventing "information leaks".
There's no known way to retrieve screenshots from the Foldit server without having the link posted in chat.
Chat screenshots remain on the Foldit server for some period of time, but they may eventually be deleted to save disk space.