Many traditional herbal medications also have significant side effects. It's not accurate or helpful to describe this as Eastern vs. Western medicine. All significant supplements and medications are bound to have side effects to some degree.
Good, evidence-based doctors don't care if the practice is Eastern or Western. If the practice or supplement has reasonable scientific evidence, a good doctor will incorporate it into their recommendations. I've had plenty of "western" doctors prescribe practices like acupuncture, yoga, meditation and traditional TCM supplements like ginger and turmeric.
The real problem with the Eastern vs. Western medicine false dichotomy is when it leads people to choose a side and stick with it. I've known a few people who suffered far too long with ineffective TCM or alternative medicine treatments before accepting proper, evidence-based treatment. The problem isn't limited to Eastern medicine, of course. For example, many depressed patients take Saint John's Wort for its herbal anti-depressant properties and are surprised to experience as many, if not more, side effects than highly targeted SSRI medications. Just because it's natural doesn't mean it's safer or more effective.