You're pretty much right on the money. For ParaCrawl[1] (which I worked on) we used fast machine translation systems that were "good enough" to translate one side of each pair to the language of the other, see whether they'd match sufficiently, and then deal with all the false positives through various filtering methods. Other datasets I know of use multilingual sentence embeddings, like LASER[2], to compute the distance between two sentences.
Both of these methods have a bootstrapping problem, but at this point in the MT for many languages we have enough data to get started. Previous iterations of ParaCrawl used things like document structure and overlap of named entities among sentences to identify matching pairs. But this is much less robust. I don't know how they solve this problem today for low-resource languages.