I ride both , somedays I feel like riding in my English saddle , somedays I feel like riding Western. It depends on what I am working on that day or what my mood is. I ride with others who ride either english or western.
Now that being said are my horses "trained" Western? Not exactly, they neck rein ( sort of) but we are still working on that. Do they have a slow western jog , well not exactly ...I can get my Dun to jog very nicely but my Paint is a work in progress (although getting better and slowing down more)
But to answer the original question it doesnt bother me at all if someone else rides Western or English .There is a lot that can be learned by both. I at one point had an instructor who taught me a little about dressage while I rode in my western saddle.
Horses and children, I often think, have a lot of the good sense there is in the world.
Josephine Demott Robinson Feed, muck, groom, ride. Repeat daily!