In the Korean language, the sir name is first followed by their given name. Perhaps the hyphen ties the given names together. As you mention “Hee-Seop Choi,” Choi is his sir name and in the Korean language would be used first.
By the way, the three most common names in Korea are Kim, Pak (usually pronounced Park by English speakers), and E (usually pronounced Lee by English speakers.)
Transliteration of Korean names is largely a personal matter. The choice of space (Jong Il), no-space (Jongil), or hyphen (Jong-il) purely cosmetic. Korean passports usually have the space, but the Korean government does not have an official policy. (At least they didn’t when we asked them about it two years ago.) Only the Korean alphabet spelling is official, so they don’t care what the Latin alphabet spelling is, within reason.
Family names can also be spelled in a variety of ways. Bak=Pak=Park, Lee=Rhee=Ee, etc. Usually a family will chose the same transliteration.
Foreign-language press should defer to whatever choice of spelling the individual makes.
I’d say to use a hyphen for the first name in English, if only to help those not familiar with Korean to differentiate between the family and given names.
In Korean they are not hyphenated at all, though they certainly could be if they wanted to do so.