In fact we just dropped all context-deduced words from exactly the same phrase and say it as 'увидимся". That is "we-who-are-talking-right-now-will-see-ourselves-(self applied verb via "-ся" particle)-in-future-at-the-same-moment'. Just 'увидим-ся' ([we] will be seen by ourselves [in future]). But it's informal. In formal speech it will evolve in something like 'hope to see you soon' / 'надеюсь скоро увидимся'. Also it can be part of more complex phrases: 'увидимся завтра', 'увидимся в кинотеатре'. And so on.

It was fun to discover english 'broken language' idiom 'long time no see' because it's literally russian greeting 'давно не виделись', that is reverse of the phrase above which can be translated as 'we didn't see ourselves a long time (and I like we do it right now)'. But AFAIK 'long time no see' has chinese origin.