I have had many students come to my dojo with various ranks gained elsewhere. I respect that they have attained these ranks and I welcome them wearing their belts in class.
Training is done (normally) with anyone, reguardless of rank, at their level. When it comes to practicing the techniques required for testing, they practice those for promotion from white belt to yellow belt (7th Kyu). On their first exam they wear a white belt and then wear their attained belt in our school there-after. Basically, they are tret as would be a visitor until their first exam. Most put on a white belt shortly after joining though.
I have had karate students come on the mat and wear their belts - coloured belts and black belts - and I have never had a problem with that, as long as the class is aware of who they are training with.
As far as Aikido black belts are concerned, if they are deserving of the standard of our school, then they remain wearing their black belt until they have passed all the gradings up to shodan. Personally I don't believe they should wear hakama, as I think it is vital to see students feet and legs.
My late Master told me 2 things relevant to this thread:
"The belt is there to hold you trousers up!"
"Show your true colour!"