I’ve had numerous encounters with badass bandits wearing amp shields at Fink’s (like two or three at the same time - keep your head down!) and sometimes in other spots too, so I can say with quite some confidence that my observations are accurate: they don’t get drained. Which makes them very dangerous as they effectively work like a Bee without the damage, capacity, and delay penalties. I made a point of at least tagging those guys regularly with some damage to keep them from having full shields (turning off the amp effect), or (preferably) mowing them down as soon as possible.
I think I’ve said this before, but rats picking them up (and most likely scavengers or LWT loaders too) and dropping them again on death makes them available to the player for picking up. When you do, they recharge your shield even if it’s not a booster shield (as expected, because it works just like that if you switch from a booster shield to something else before picking up one of your own boosters).
Not sure if the rats receive the booster’s effect. I’d guess not, because they store the booster instead of using it - I’ve seen them pick up shields, gain a shield bar and start charging it (i.e. they do use found shields), then drop the shield again on death; but boosters are a kind of item that usually vanishes when used. They might bypass that like the amp drain though, so that’s something to check.
Pete’s bar would probably be the ideal testing ground for this since you can have enemies with booster shields and rats in the same place, and they won’t even attack you initially (for easy observation). The booster shields of course need to be damaged to trigger boosters, but maybe Scorn can do that without alerting them. The small amount of damage it deals to the rat’s shield (making it hard to tell the difference between full and slightly damaged shield after it picks up a booster) can be offset by dropping down from UVHM to Normal Mode. If enemies with booster shields refuse to show up, donning one yourself and dealing self-damage somehow might work too.
A player’s boosters can be stolen too. I don’t think it’s practical to abuse this to direct scavenging enemies due to the inherent randomness of booster shield activation (even when the boosters are the WTF’s IED boosters for some free damage) but the potential is there. This would probably be the best use for that useless thing called Pot O’ Gold though, since the one character who can really benefit from picking up such tiny amounts of money is in another game (Jack the Doppleganger with his skill Money is Power).