As animenerd mentioned, I believe hot glue can melt styrofoam, though I haven't really tried. To fuse foam together, you can use white glue, though it can take awhile to dry, or 2-part epoxy, or gorilla glue, which must be used sparingly since it foams up. You can also paint the surfaces in cheap acrylic paint and then fuse them with spray-glue or contact cement.
The bond doesn't really have to be that strong, it just needs to hold together while you're carving it and applying the paper mache. After that, yes, the paper mache will hold everything together.