where does the sewer go if you live on an island?

Some islands would still have a sewage treatment plant -- they wouldn't have to have pipes get to the mainland, just the local plant.

A truly private island would probably have a septic tank, just like more isolated homes on the mainland. And I suppose a few with maybe only one or two residences would also possibly just dump into the ocean, or else have things shipped out for disposal much like they need supplies shipped in... maybe using a system sort of like you find in an airplane or RV.

Um, pretty much all sewage goes eventually into the ocean. It is treated in a treatment plant, then dischaged into a river, which then flows into the ocean. On an island, you just skip the river part.

Some homes have septic systems, which house the 'stuff' in a tank underground, where it is treated with bacteria that break down the 'stuff' and then releases *relatively* clean water.

There is no sewer. Most likely a septic tank.

If it's like the majority of the places in the world, the raw sewerage will go directly into the ocean.