What byproducts are there?

I'm not aware of any. In relation to plastic degradation, typically the fungi break down the chemical bonds of the molecules with the help of enzymes.

How do we prevent it spreading to plastic we don’t want decomposing?

I remember one research from europe (I can't find it for some reason) that they were collecting the plastic first and then in a controlled environment they introduced the plastic-eating fungus. It's not that the scientists just threw the fungi in the ocean.