Psipsina - dogs are trainable, but, it's not as straightforward as just training them not to chase cats.
They don't generalise, each behaviour you teach has to be practised in different circumstances and settings before it can be considered anything like solid.
You can have a nice solid leave for dropped food, rabbits on a walk...that doesn't mean that they'll also apply that command to a cat in their garden.
Not all dogs chase cats, not all dogs that chase cats would hurt them if they caught up with them, dogs benefit just as much as any other animal from time outside where they're not under close control...that is what dog owners use secure gardens for.
If you own a cat, let it out without putting up a pen or cat proof fencing you accept, sadly that there is a chance that they may be killed by a dog, a fox or far more likely a car.
I own cats an a dog, they don't need to be on separate islands.
None of that excuses a dog's owner from being uncaring if the worst happens and they absolutely should let the owner know.