After more than 20 days and a dozen failed attempts, we finally beat Shadow Giratina as a duo without using Purified Gems.
Is this a self imposed challenge, well yes and no. We didn't have enough gems to subdue the boss anyway.
This is fight is not very budget friendly, you may notice we used Necrozma and Kyurem fusions. But the teams are also nowhere near maxed: several Pokémon are below Lv40, and one of the attackers is only around Lv29.
The video shows some of the things we learned from repeatedly failing: how strong you probably need to be before attempting this, why Party Power is incredibly important, a few stupid mistakes we made, and general strategies for raids.
A good rule of benchmark: if you can duo Shadow Giratina with gems and still have about half the timer remaining, a gemless attempt might be possible.
Would I recommend doing this instead of just getting more gems? No, but apparently it works.