Chapters could be written on this.
The short answer is that the
standard practice was for each Emperor to strike coins for all of the others whom they recognized as official co-emperors.
Thus "splits" like under
Constantine's family, Valentinian's family,
Theodosius' family, etc. were official and were more like the division of the Empire under
Diocletian's
tetrarchy than an actual split into independent polities. This was generally true whether each
part had a little independence or a lot of independence - as long as they recognized each other.
By contrast, usurpers and rebels were usually not recognized by existing emperors, though sometimes the usurper or rebel would strike coins for the official emperor thus "recognizing" them and signalling that they were seeking to in turn be recognized as a legitimate co-emperor.
Though the usual practice was for emperors to strike coins for each other there were
still some more subtle politics (court intrigue) that were played. For example, one emperor might not strike coins for all of another emperor's sons, thus coins for some caesars at some times only appear in the territory of daddy, not others. Or they might signal a lower status for others with such obscure things like un-broken legends or bare heads.....
SC