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Well it would mean that "I am bored without you"... which sounds kinda strange in English too...

Not necessarily... If you are talking to you spouse and are talking about feelings and stuff, and say "скучаю" in a particular tender voice with at least three resonant frequencies, then it would...

Штандартенфюрер СС фон Штирлиц, вы уж медлены. :D это говорит человек внизу налево.

я скучаю по тебе (ya skuchayu po tebye) (I miss you)

or simply

скучаю (skuchayu) (I am missing... meaning implied)

I hope Solzhenitsyn misses you too!