There are many varieties of Jasmine that grow fairly well here. With the exception of the yellow vining Jasmine which can take full sun all year around, the rest need protection from the hot afternoon sun most of the time. So you need to plant them where you get morning sun and afternoon shade for best results.
The popular and available Jasmine for the Phoenix metro area are:
Poly pink jasmine, vinelike, extremely fragrant white/pink flowers, part shade.
Asian Jasmine, shrublike, with white lightly fragrant flowers, part shade.
Star jasmine, shrublike, with white lightly fragrant flowers, part shade.
Arabian jasmine, shrublike, with white slightly fragrant flowers, part shade.
Yellow jasmine, vinelike, with slightly fragrant yellow flowers, full or part sun.
Out of the 23 or so jasmine varieties, the above listed are the most popular (sold by most garden centers) for the Phoenix area, and they do quite well with part shade, moderate watering, moderate fertilization. Bees are attracted to these plants, so be careful where you plant them.