What does Anderson Island do with it’s trash? Do they have a dump?
It looks like there’s a garbage transfer facility.
Apparently the same sanitation company that does household pickup for other small communities in the area does them too: http://www.co.pierce.wa.us/pc/services/home/environ/waste/recycle/garbage/lemay.htm
I’d guess they just drive the garbage trucks onto the ferry and dispose of it on the mainland wherever. If you’re really that curious you could try calling or e-mailing them.