Funny. To me, it's such a strange mindset not to work together. I know that's the norm, but I can't understand the motivation to wall everyone else out.
I was very lucky coming up that my instructor was open to collaboration and cross training. Our TKD club would get together with a local Wado Kai Karate club for sparring nights from time to time (only for senior colored or black belts). Both head instructors monitored the floor, but there was never a problem (the joys of mutual respect).
I currently am teaching in a very small town and have built a solid relationship with the few other local club owners (TKD for me, Judo, and Kickboxing/BJJ). We've all referred potential students to each other (when their needs would best be met at the other club). We get together for community events and tomorrow will mark our 1st foray into a weekly instructor's cross training class (we're starting with the Judo club taking lead).
There's so much we could all learn from each other if we check our pride at the door. IMO other martial arts are not my competition (soccer, dance, or baseball on the other hand...)