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 mostly-completed Pokédex (minus some Gen 2 and all Gen 3 Pokémon) on FireRed is described here. This page will focus more on requirements specific to a living dex using only FireRed and LeafGreen.
You will need access to at minimum three separate FireRed or LeafGreen save files in order to obtain all starters and legendary beasts. Each file will need to progess as far as defeating the Elite Four and delivering the Sapphire to Celio on One Island to unlock the legendary beast.
This is just to note that if you are attempting to complete a living dex on both FireRed and LeafGreen simultaneously, you have the ability to breed almost any Pokémon, which can assist in preparing version exclusive and mutually exclusive Pokémon for trades. The Pokémon that cannot be bred are Legendary and Mythical Pokémon, Baby Pokémon, Nidorina, Nidoqueen, Nidorino, Nidoking, and Unown.
This is just to note that if you are attempting to complete a living dex on both FireRed and LeafGreen simultaneously, you will need at minimum six save files, in order to ensure that each copy of FireRed and LeafGreen gets access to each of the three Legendary Beasts, since they cannot be bred.