You are correct about dumping "everything" overboard back in the day. We dumped paint cans, paint brushes, clothes, oily rags, office chairs, desks, file cabinets, old tool chests, drop tanks, aircraft canopies, mops, mop buckets, you name it... it went over the side. Including tons upon tons of plastics and non-biodegradeables.
Nowadays they have to sort the trash and pick out all plastics and oily products. The ships have big oven type lookin' machines that take all the plastics and melt them down into big "pucks" for storage until the ship comes ashore and can offload them.
However... the ships still dump all paper, metals, and anything deemed by the US Navy to be biodegradeable over the side. All the floor stripper, waxes, and cleaners are now biodegradeable so they can be chucked over the side.
So when the trash dumping detail is set... they spread huge tarps in the hangar bay, and each division has to send their share of people down to help sort trash. Also good detail for the restricted sailors as well. After the trash is sorted, the stuff that is to go over the side, is rebagged in huge biodegradeable paper bags and hucked over the fantail... This is overseen by the XO (Executive Officer), who will be on scene with the Master At Arms (Ship's Police) to inspect what goes over the side. It is an offense punishable by the UCMJ (Uniform Code of Military Justice) for tossing any non-biodegradeable item over the side. It is taken very, very seriously. As it should be.
Everything that is plastic or styrofoam is sent to the plastic ovens... anything else that cannot be sent over the side, is stored in the aft hangar bay in huge tri-wall cardboard containers for offload when the ship comes in to port.
Times have changed the way the US Navy does it... but we still dump trash over the side...
Gary