You can’t use your “Precious Cards from Beyond” when your opponent offers 2 monsters as a Tribute, only when you do.
“Precious Cards from Beyond” is only for a Tribute Summon or Set, not for a Ritual Summon or Special Summon like “Lava Golem”.
If you Tribute Summon “Blue-Eyes White Dragon” for 1 Tribute using “Cost Down”, you can’t activate “Precious Cards from Beyond”’s effect.
If you have multiple “Precious Cards from Beyond”, you can activate each of their effects when you Tribute Summon or Set with 2 or more Tributes.
If you Tribute Summon or Set with 2 monsters, and 1 of them was your opponent’s monster that you used with “Change of Heart” or “Soul Exchange”, you can activate “Precious Cards from Beyond”.
Netrep Q&As:
Q: Can the effect of “Precious Cards from Beyond” be used when summoning “Lava Golem” to the opponent’s side of the field?
A: The effect of “Lava Golem” is not a Tribute Summon. It will not work with the effect of “Precious Cards from Beyond.” Tribute Summons are Normal Summons of 5+ Level Star monsters by offering Tributes. Curtis Schultz, 02/04/2004