This word is a Scots colloquial term, though not a common one in print. It means a place to sit out in, a summerhouse or gazebo, from sit plus oot (a Scots pronunciation of out) plus the noun ending –erie of French origin that’s familiar from words like menagerie and rotisserie.
In the flickering light from a distant candle my partner and I sat in a “sitooterie” to partake of tea, pie and cakes.
Motherwell Times, 10 Mar. 1933.
English newspaper readers suddenly started to see this word during the summer of 2000 because it was applied to an art exhibition in the historic landscaped gardens of Belsay House in Northumberland, near Newcastle upon Tyne. A dozen designers and architects were each given a budget and invited to interpret the idea of a sitooterie as a meditation on the perception of landscape. This resulted in intriguing structures, some practical, some more like follies. The exhibition had the minor consequential effect of turning sitooterie for a brief period into part of the English — as opposed to the Scots — tongue. It has since vanished again.
Several Scottish subscribers have remarked that the word used to have a rather different meaning — a secluded corner where you could take your partner during a dance. It would seem that the word has either shifted sense, or the exhibition organisers have extended its meaning.