ardly: I only speak one language and I don't have a formal understanding of it's grammatical rules. I am sure that is a bad thing. I suppose though as a non-native a lot depends on what level of study you are undertaking. If you are studying at University level being tested on a formal understanding of the rules probably makes sense.
It is the language that dictates the grammar and not the other way. A native is not (necessarily) required to learn the 'inherent rules' that are embedded within the language as it (the Language) is his mother tongue. On the other hand, a non-native (like me and others) acquires the 'writing style' of the Foreign Language through hard exercise of the 'syntax and semantic' rules of the language. A native can have much better control over the writing/speaking style of his mother tongue having known the mechanics of the working principles of the grammatical rules/conventions of the language.