I think it depends. If my student went out and got a Dan rank with another school, I would consider them to have a Dan rank with that school. When they came to my school, I would expect them to wear the rank that they earned under me. I would respect their decision to test, and that they have a rank in another school. They would just need to earn the same rank with me to wear it in my school. Full disclosure, I have my Nidan from the INKKS, and certainly didn't have any communication with my original sensei about it. There's a lot of complication behind that, but even if we were still communicating, I wouldn't mind if he still views me as a Shodan. Similarly, with my current dojo, they know that I am a Nidan, but I wear a Sankyu when I am at that school because that is the rank that I achieved with them.