The answer should be plain and simple : Yes, the jinchuuriki gains the nature of his or her bijuu. It doesn't change the jinchuuriki's nature, instead it just adds another affinity. If you had the water affinity, and you have a fire bijuu, then now you can use water and fire as well as the combination of water+fire together in a technique.
Naruto is naturally a wind element shinobi. With the kyuubi inside of him, he can use kyuubi attacks (yin/yang).
It seems like a general rule that once you have something (chakra or body part) of a different nature inside your body, you are able to use attacks of that different nature too. Take Kakuzu as an example. He has multiple affinities due to the fact he stole other people's hearts and stuffed them in his own body.
Having multiple affinities does not necessarily mean they will mix and you can't use them separately. Take Yamato for example, as his affinity is water and earth, which also turns into wood. He can use wood techniques (the combination of the 2), but then also use them individually as he once demonstrated by raising the earth and then creating a waterfall.