All Koreans are required to learn Chinese characters. I hated it when I was young and I lost almost all of it. But even if you don't know how to read and write the actual Chinese characters, you can still retain the meaning behind it.
For example, the word consider the word 무료 (無料) (Moo Ryo) ("No" Charge/Free). Now, I forgot how to write that in Chinese - I just looked it up and copied and pasted Chinese characters. But I know word Korean word 'Moo' has several meanings, and one of them is Moo as in 無 in Chinese, meaning "nothing," "no," "non-existing," etc. Even though I don't know the actual Chinese character, I know many Korean words that have that word in it - 무자비 ("Moo" Jabi - "No" Mercy or merciless/ruthless), 무죄 ("Moo" Jwe - "Not" guilty/acquitted) , 무책임 ("Moo" Chaekim - "Not" responsible/irresponsible), 무지 (Moo Ji - "No" Intellect/ignorant), 무한도전 ("Moo" Han Do Jun) - Unlimited ("No" Limitation) Challenge or Infinite Challenge), etc.
So you can see that if you know a lot of Korean words and know the meaning behind the root character, you can tell the meaning of the word, even without knowing the actual Chinese character. The same thing with the second character of 무료 Moo "Ryo." The word 료 or 요 (Ryo or Yo, depending on where this character is positioned in a word), means "fee" or "charge" and even if you don't know how to write that in Chinese, you can also know the meaning of 요금 (Yo Geum), which is means "charge money" or "fee."
Here's another example. 무관 (Moo Gwan) could have several meanings. It could mean irrelevant/ "No relevance" (無關), A military officer position in the olden days (武官), Not holding a position in Court) (無官), or it could also mean No Crown (for the king) (無冠). They are all spelled and pronounced the same way in Korean, and most people can tell the meaning by the context. But if you knew the Chinese root characters behind it, the meaning is really clear and there is no question behind it.
I've seen a lot of subbers making mistakes when translating and using another meaning of the same word. So it certainly helps if you knew Chinese. But like I said, if you are fluent in Korean, you can pretty much get by without knowing Chinese.