The invented languages of twins, sometimes called 'twinspeak' are also known as cryptophasia or idioglossia. Peter Bakker of Aarhus University (Denmark) has published on this topic (abstract available here). According to Bakker twin languages are very common, occurring in about 40% of all twins, but soon disappear.
Regarding the structure of these languages, Bakker writes:
In all cases known, the language consists of onomatopoeic expressions,
some invented words, but for the greatest part of words from the adult
language adopted to the constrained phonological possibilities of
young children. These words being hardly recognizable, the language
may turn out to be completely unintelligible to speakers of the model
languages, but they resemble each other in that they lack morphology
and that word order is based on pragmatic principles such as saliency
and the semantic scope of words. Neither the structure of the
languages nor its emergence can be explained by other than situational
factors.
The Bakker article seems to be hard to obtain. Further reading may be found in the bibliography of the subject Bernard Comrie posted to LinguistList in 1998.