Q. “Do these two sets partitioning a boundary have a common name?”
A. Yes. They are the rims of $A$ and its complement.
Q. “Who studied them?”
A. Kar-Ping Shum
References
-------------added 7 Oct 2017-----------------
Should have also mentioned: early authors including Kuratowski used the term border to describe this operation. It seems to have faded away at some point (probably before Shum and Yip's 1975 paper). I prefer rim, because in English, the words “border” and “boundary” generally apply to multiple objects at once (the ones being separated by the border or boundary), whereas something's “rim” applies to itself only.