Does anyone know if the languages all live in the same 300-dimensional space, or are they each trained independently? (i.e. do words and their translations have similar vectors?)
Models are trained independently for each language. So unfortunately, you cannot directly compare words from different languages using these vectors.
If you have a bilingual dictionary, you might try to learn a linear mapping from one language to the other (e.g. see https://arxiv.org/abs/1309.4168 for this approach).
If you have a bilingual dictionary, you might try to learn a linear mapping from one language to the other (e.g. see https://arxiv.org/abs/1309.4168 for this approach).