Converting between strings and array of bytes?

But if I do it on the right side, and send it like that, the modules will get loaded by the IDE and be save()ed

Yes - well, they'll be loaded by the Web IDE and will be save()d when you type save().

If there are two versions of a module with the same name, the one on the website, and the one on the SD card which one will be used?

The one on the website (if you use the right-hand side). You can force loading from SD by doing:

var foo = "mymodule";
require(foo);

At what point in the process would the IDE tell the Espruino to load that?

When you click Send to Espruino it prepends the contents of the module to the code that you wrote.

for example, would this force it to load off the SD card, or does it do it at the end?

No, that would force a load off the SD card.

It's a little disconcerting how it isn't made clear when the IDE is summoning a module

Well, it's meant to be virtually invisible. It might make more sense if it wrote some description onto the console I guess...

is there a location on the local filesystem that the IDE will check before it goes online?

No, but you can change the URL used for modules in settings, and can set it to a local webserver.