A libretro implementation should be compiled into a dynamically loadable executable (.dll/.so/.dylib) or a static library (.a/.lib) that exports all the functions outlined in libretro.h.
What programming languages satisfy the above requirements? I know that C and C++ can use the library directly. I also know that there are Rust bindings. Are there other languages that work, though?
Yeah, pretty much anything that can work with C should be fine.
In the libretro-samples repository, there are also examples about freebasic and pascal 
For Rust, there is for example this project Unicorn Console which has libretro backend.
I also have a python sample here https://github.com/bparker06/button_test_py that uses C to set everything up normally and then calls retro_run() from a python script.
That is an interesting way of working with Python. How would that work with a Libretro frontend? What about the fact that Python is an interpreted language?