How does the Sleeping Giant work?
You won't be notified at all. The bonuses vary wildly from 3x damage to 2x fire rate. But some aren't buffs. A few people have reported a drop in damage. I can't say I have ever had this happen personally. The way the gun works is every time you reload it you get a new random bonus until you reload again. As far as I can tell there is no way to predict what bonus it will take.
On a completely different note, this thing gives a good bonus paired with the Moze's "Some for the road" perk, which alone is quite powerful.