Neither one sends the battle. In Pokémon Red and Blue's code (the English cartridges, 1998), the two Game Boys swap teams and a list of ten random numbers when you sit down in the Colosseum. After that, each turn, the only thing that crosses the cable is each player's pick: four bits, a move slot or a switch, while your screen says "Waiting...!". Then each Game Boy plays the whole turn itself, and every roll (critical hits, misses, the damage spread) is the next number on that shared list, so both screens agree.

Each Game Boy plays its owner as the player and the friend as the enemy, and the two halves of the code don't quite match. When Psywave rolls a zero, the player's half throws it out and rolls again; the enemy's half keeps it and does nothing. From then on the two Game Boys read the list at different places and play two different battles. (The list: both keep the ten numbers of the Game Boy that clocks the cable; only nine are ever read, and after the ninth each one becomes five times itself plus one.)

Satoshi Tajiri told TIME in 1999 that the cable is what got him interested: "I thought of actual living organisms moving back and forth across the cable." And: "It's an intricate style of communication. Almost subtle."

The handhelds, the two sides, the screens and every number on them are our own simplified version: a port of the game's rolls with our own numbers and lists, shown slowed down after the turn. Not affiliated with or endorsed by Nintendo, Game Freak, Creatures or The Pokémon Company.

Did a link battle ever end differently on your friend's screen?

Sources: the game's code, read in the pret/pokered disassembly of Pokémon Red and Blue (cable_club.asm; core.asm: BattleRandom, LinkBattleExchangeData, CriticalHitTest, MoveHitTest and the two Psywave damage loops); Tim Larimer, "The Ultimate Game Freak", TIME Asia, 22 November 1999.
Sounds: our own, made in code.