Add them as a Straight Pool Deluxe opponent, not as a local player, and every game between you shows up on both iPads.
There are two ways to put someone in your Players list, and they work very differently.
For opponents who don’t have the app
Typed in by hand with the plain + button, or by typing a name into a game.
For opponents who also have Straight Pool Deluxe
Added with the blue + SPD button.
If the person you play also has Straight Pool Deluxe, add them with + SPD so the games between you sync to both iPads. Here’s how, step by step.
Rule of thumb: before you add a new player, ask them, “Do you have Straight Pool Deluxe?” If they do, ask for their username or the email they registered with, and use + SPD.
Open Settings → Players and look at the badge next to each name:

Score a game against a connected opponent and it appears in their Games list too. They don’t have to do anything.

No problem, nothing is lost. Here’s what to do with the local player and your old games.