No problem with sharing similar training. There's no sense in having 2 tanker schools, for example.
I would object to "joint" basic training. I think Marine or Army basic, e.g., is entirely different than navy or AF.
There's no overcomable reason that some functions cannot be joint. Chaplain, lawyer, nurse, psych, and doctor can be joint. It'll be harder with medical. There are differences in the type of wounds one would receive in a ground unit as opposed to an air unit. You might want "specialists" in the medical field.
I'm a retired Chaplain, by the way.
I can see a purple suit for Chaplain, medical, legal, counselor, etc. I can even see them being outside the rank structure.