It's not like there isn't a phrase for it in Russian (Po tebe skuchayu).
Russians definitely feel it. It's the feeling that you're alone, but not just that you're alone, that you're alone specifically because you are lacking someone else, even if you're with other friends or family, that someone specific isn't there and that there is something missing.
In the early part of most relationships you'll find that you're thinking of the other person at odd times, and you might get the sensation that they're seeping into the empty spaces in your mind and filling them up when you're not preoccupied with someone else. Most of the time that sensation fades after a while, but in the best relationships, it stays, and that is when you are missing someone, when the memories of them that are filling up the gaps in your thoughts are so strong that you feel an almost physical reaction to them (or maybe you do have a physical reaction to some of those memories, the memory of a touch or something).
I think a lot of people say "I miss you" without meaning it much, just as some people when in a relationship will say "I love you" without feeling it. To me they're two sides of the same coin. When you love someone, it feels like they fill you up and complete you, and when you're in love, they feel the same way about you. If you're apart and not occupied with something else, you'll feel that something isn't right or that you want something else, and you'll realize that it is your significant other.
I can definitely say that this feeling is stronger in my current relationship than it ever has been before, and it hasn't worn off even though it's been a good while yet. To me that's a sign that there is something truly different about this one