A "living dex" is a fan designation for when a player completes the Pokédex and has one of each species in the PC or the party at the same time. It's often considered to be much more difficult and time-consuming than a regular Pokédex because it often involves, for instance, playing the same game multiple times to get multiple copies of a Legendary Pokémon.
The method by which you can get a completed Pokédex on Shining Pearl is described here. This page will focus more on requirements specific to a living dex.
Only one save file of each of Brilliant Diamond and Shining Pearl is required, since starter Pokémon are available in the Underground and therefore are not mutually exclusive selections.
This is just to note that if you are attempting to complete a living dex on both Brilliant Diamond and Shining Pearl simultaneously, you will need at minimum four save files (two on each game), in order to ensure that each Brilliant Diamond and Shining Pearl gets access to Dialga, Palkia, Articuno, Zapdos, Moltres, Lugia, Raikou, Entei, Suicune, and Ho-Oh.