It’s not ambiguous in Swedish, it refers to the actions here. Otherwise you’d use ”båda” if you were talking about the people. ”Båda” is here a pronoun whereas ”både…och” are two conjunctions that go together.
- Vi är båda läkare. (We are both doctors.)
- Både han och jag är läkare. (Both he and I are doctors.
- Båda husen ligger vid sjön. (Both houses are situated by the lake.)
- Att leva med honom är både himmel och helvete. (Living with him is both heaven and hell.)
I read it, but I don't see how it is not the meaning of this sentence. Vi både skrattar och gråter: We [us, both, all] are laughing and crying [both]. Would the meaning change at all saying vi båda skrattar och gråter? Would it not simply be grammatically incorrect, but mean the same thing? Or is the insinuation that the 'we' could be more than two, and the use of båda would make it mean specifically only two people? Det är både roligt och svårt att lära sig svenska ;)
OK, perhaps it will help to change the sentence so the English translation loses its ambiguity. So let's imagine it says "Jag både skrattar och gråter" - "I am both laughing and crying". Now it's clear even in English that the 'both' refers to the verbs. The point is that in Swedish that was already clear, because they use a different form of the word. In fact, I think (someone who knows about this please correct me!) 'både' is a conjunction and 'båda' is a pronoun. In English we use the same word for two different functions but the Swedes don't. Your sentence is grammatically correct in English but it's not a correct translation from the Swedish, because you've changed the function of 'both'. Possibly I've just made it even less clear...
You're right about the conjunction/pronoun thing and everything else. I wrote some more about this recently here: https://www.duolingo.com/comment/8387413$comment_id=15263308
From the Swedish sentence, we don't know how many people are involved here. But we do know that only two actions are involved: laughing and crying. In the Swedish sentence here, it is to those two actions that the word både refers.
In other words, the structure of the sentence is NOT [We both of us] laugh and cry. Rather, the structure is We [both laugh and cry].