Abstract Recent research in cross-lingual word embeddings has almost exclusively focused on of- fline methods, which independently train word embeddings in different languages and map them to a shared space through linear transformations. While several authors have questioned the underlying isomorphism assumption, which states that word embeddings in different languages have approximately the same structure, it is not clear whether this is an inherent limitation of mapping approaches or a more general issue when learning crosslingual embeddings. So as to answer this question, we experiment with parallel corpora, which allows us to compare offline mapping to an extension of skip-gram that jointly learns both embedding spaces. We observe that, under these ideal conditions, joint learning yields to more isomorphic embeddings, is less sensitive to hubness, and obtains stronger results in bilingual lexicon induction. We thus conclude that current mapping methods do have strong limitations, calling for further research to jointly learn cross-lingual embeddings with a weaker cross-lingual signal.