If I have 5 cards in my Lost Zone and my opponent has 5 cards in their Lost Zone, can I use Sableye's "Lost Mine" or Giratina VSTAR's "Star Requiem" attack or Fantina?
No. Each player has a separate Lost Zone above their Prize Cards that they put cards into. You yourself must have 10 cards in your Lost Zone in order to do either of those attacks or to play Fantina.
Source: Lost Origin FAQ; TPCi Rules Team (2022-09-08)

Relevant cards

Note: All applicable cards may not be listed below.
SWSH11_70
SWSH11_70
SWSH11_131
SWSH11_131
SWSH11_157
SWSH11_157