The reapers in Eorzea watch out for Garleans who have fled the empire after growing sick of its evil. They take care of the imperial deserters in Thanalan and regard them as family
As for Garlemald proper I'm not how far along you are in the Reaper questline, but
They stay hidden and exist to strike down corruption in political and social leaders, similar to dark knights in Ishgard. Drusella's grandfather tried to assassinate Emet in the empire's early days, though obviously didn't succeed.
I would imagine most Garleans probably don't even know they exist, that they're a rumor at most. And given the almost cultlike mentality Garleans hold, the reapers'actions against Garlean leaders probably isn't seen in a good light.