"He is my boyfriend, I am his girlfriend."
Translation:Han er min kæreste, jeg er hans kæreste.
It's nice that the word is non-gender-specific, now I can speak about my (theoretical) kæreste without revealing that I am gay :D
These are 2 independent clauses that require the conjunction "and" between them for a proper translation into English. I am assuming that the Danish as written is correct without the conjunction "og".