When moving files between two external drives, are they temporarily written to the internal hdd of the computer?

If you have a copy such as this, or its GUI equivalent,

cp -a /media/external/disk1/. /media/external/disk2/

the data is read from the first disk's filesystem and written directly to the second. There is no intermediate write to another storage location. If you are seeing slow speeds it may be that the two disks are sharing the same USB controller and contending for access to the bus.

Anything more than that and you will have to provide further details, such as the make/model of computer, its bus topology, etc.

There is no concept of internal and external: The operating system knows not of the box, and what is in or out of it. The only file-system that is special (but only a little bit) is the root file-system, and this can be mounted read-only. You can still do almost anything with a read-only root file-system (but not change the content of the root file-system).