When it comes to the dumbbell snatch, you can either pick up the dumbbell with a vertical grip or a horizontal grip.
The vertical grip will mimic more of a kettlebell swing as the path of the dumbbell will loop around the body. This is the preferred method of most athletes because you can recruit more of your posterior chain (hamstrings, glutes, and low back). Additionally, when the dumbell is overhead, a vertical grip feels more natural as the weight must be placed directly over your shoulder.
The horizontal grip mimics the barbell snatch better than the vertical grip. Two of the major benefits to a horizontal grip is that you can keep the dumbbell closer to the body and you can recruit the same sequence of muscles in the second and third pull of a barbell snatch.
The horizontal grip in the dumbell snatch mimics more of a barbell snatch. The vertical grip mimics more of an american kettlebell swing. Either way is correct and preference should be given to whichever feels more natural.
However, if you were to do a double dumbbell snatch, I would recommend doing only the vertical grip. Placing the dumbbells horizontally between your legs in a dumbbell snatch would require that you start in a sumo stance–something that wouldn’t be optimal to driving weight overhead.