Either a glove or mitt can be worn at any position in ASA, USSSA, NFHS (high school) and NSA fastpitch softball. So, yes, your pitcher may wear a first base mitt- and your shortstop could wear a catcher's mitt if she wanted to!
This used to be different for some sanctioning bodies, where the use of a mitt was confined to the catcher and first baseman only. Over the past few years many associations have modified this rule to allow any type glove or mitt at any position. The reasoning behind this has been that a mitt offers no inherent advantage at other positions and that organizations do not want to limit participation for lack of a specialized piece of equipment.
While the above four sanctioning bodies are the ones I most commonly deal with, I would be willing to bet that there are still sanctioning bodies out there that do not allow mitts at any position other than F2 and F3. I believe that Little League, and perhaps PONY, still have this restriction, but I don't have their rule books handy.
Beyond fastpitch, strangely USSSA slow pitch still limits the use of mitts to F2 and F3, as do most baseball organizations. High school baseball is the only one I'm aware of that allows mitts at any position, the same as high school softball does.