Unfortunately, I was not able to find empirical literature on this question, as it does not appear to have been studied much - for example:
References to dreams in the multiple personality literature are rare
and tend to be brief.
So we are limited to clinical reports. In 1994, Deirdre Barrett collected 3rd-hand accounts of dreams from 23 DID (MPD) patients. The study method that she used is not very good quality however, so any conclusions would be premature, but it does provide some possible hypotheses as to the nature of these patients' dreams, as summarized in this table:
In particular, alters appear as characters in dreams (4) of more than half of the patients in this survey. Item (5) lists a few patients reporting dreams experienced from multiple perspectives (ie, multiple alters) simultaneously:
... for these 26% of patients with dreams in category 5, two or more
personalities reported dreaming at the same time and experiencing each
other as characters.
Another interesting item (9) lists patients who reported experiencing multiple different dreams by different alters. Note that some of the dream reports (2) were not actually dreams, but were perceived as such by the alter recounting them.
More case reports are reviewed in another paper, and Barrett has written extensively on this subject. She summarizes:
These included memories of childhood traumas and content of recent
fugue episodes. A few of the dreams were undisguised recreations of
actual episodes, especially for recent repressed events.
No characteristic of the dreams of dissociative disordered patients in
this survey is so distinctive as to never be found in those of other
Empirical data (based on mentation) would be more reliable for drawing conclusions. Barrett and others have hypothesized that rather than alters appearing in dreams, characters in dreams may be used as inspiration (prototypes) for dissociated identities.