You are welcome. By the way, I'm also in the course of learning basic Japanese so my explanation
may not be perfect.
Yes, something already implied (clearly) can be left out of the sentence, and I think this is true for all Japanese sentences.
The same is true for the Korean language, too (I'm a South Korean), which has a very, very similar sentence structure with the Japanese language (In fact, there are numerous South Korean words that are exactly the same with the Japanese language because we both use the Chinese characters!). Okay, this is off topic.
Anyway, yes, if you want to ask
"Where do you work?" to someone, you can just say
"どこに働いてますか。" instead of
"(あなたは)どこに働いてますか。" because it is clear to the listener that you are talking to him or her (that is, him or her is the subject you are talking to).
By the way, Google Translate is not quite accurate with its translations - although it's indeed am intelligent Mechanical Turk. It can be even more inaccurate when you are translating from one language to another language both of which are of
different linguistic roots (i.e. Translating from Japanese to English (these two have different linguistic roots) is not as accurate as translating from Japanese to Korean (these two have similar linguistic roots such as the using of Chinese characters (Kanji); both countries were greatly influenced by Chinese Confucianism a lot in the past.). Just saying that depending too much on Google Translate may not be a good idea, though I does give you a vague implication of what the source text is about.
Hope this helps.