It is now established that modern neural language models can be successfully trained on multiple languages simultaneously without changes to the underlying architecture, providing an easy way to adapt a variety of NLP models to low-resource languages. But what kind of knowledge is really shared among languages within these models? Does multilingual training mostly lead to an alignment of the lexical representation spaces or does it also enable the sharing of purely grammatical knowledge? In this paper we dissect different forms of cross-lingual transfer and look for its most determining factors, using a variety of models and probing tasks. We find that exposing our LMs to a related language does not always increase grammatical knowledge in the target language, and that optimal conditions for lexical-semantic transfer may not be optimal for syntactic transfer.