I assume you're talking about learning a second language after you've already established a dominant first language, referred to as an L1. Research by [Kroll & Stewart, 1994](http://www.pitt.edu/~perfetti/PDF/Kroll%20&%20Stewart.pdf
) indicates that the underlying concept for all versions of a given word is stored as a unit, and that retrieval of that particular conceptual unit facilitates word memory in all the languages known by an individual. This word memory is usually funneled through a person's L1. In other words, finding a word in a second, third, etc. language often occurs through accessing the specific word for a concept in a person's L1 then translating it to the other language, even if this process isn't done through conscious effort. A similar thing happens with grammar: a person with an L1 will usually relate grammar in other languages to the corollary grammatical structures in the L1. Noam Chomsky's early work argues for a universal grammar, which asserts that humans possess brain functions specifically adapted for use in language, and that underlie all languages. His theory leans toward the "additive" linguistic model proposed in your question. However, new languages do require the brain to develop new neuronal pathways associated with that language. In that sense, the second language is indeed "separate" because it requires different neuronal pathways than those used for the L1 to be functional.
More simply, the answer to your question is both. A lot of the semantics (i.e. word meanings) and grammar for each language are stored separately, but they're constantly interacting, too. The new language is not held in some disassociated network: it interacts with the other languages you know all the time. Nonetheless, it does require a "different" set of pathways to function. It might help to think of the two languages as parallel highways (they basically go to the same places) that have a lot of interchanges between them.
As far as the question from /u/Sir_Spaniard is concerned: I'm not aware of any studies like you're looking for and I'm too lazy to research it right now, but I can tell you that personality is largely governed by the frontal cortex of the brain. Maybe that helps you research it yourself. Also, it's a serious area of contention in the linguistic community whether there actually are personality changes that can be attributed to the use of different languages.
Add on question: I've come to understand that we can formulate different personalities when we speak different languages, what parts of the brain are responsible for this and have there been any studies (for example, looking at different parts of the brain via an MRI while the person jumps around through different conversations in different languages)