Shadowverse: Starforged Legends

Are we playing with decklists?

No. It’s too cumbersome and not always possible to verify.

If not, do we have the right to leave a private match to, for example, slightly change the deck to suit the matchup?

No. You play with what you picked (provided the choice doesn’t violate other rules). Consider this also part of the strategy – to predict what the opponent will play (which craft/which deck).