Doesn't sound that fancy... Orbital calculations are hard maths, but when you've written code to do it, that code can run anywhere, including on the satellite. Very little compute power is required.

Also, satellites are in direct view of one another and can easily receive each others signals. A rather simple software radio could receive the signals from a basic omnidirectional antenna. The only challenge is subtracting out the very strong local signal before digitising, but considering the very strong coding gain in GPS, it should be doable.