In Dragon Ball Z, after fusion, who controls the mind and the body? Is it both or a single person?
There are several types of fusion. For the majority of these, the more dominant/stronger individual controls the mind and body.
For Piccolo and Nail, they did the Namekian Fusion, Piccolo was the more dominant. There was also Cell and the Androids. Cell fused with them by a type of fusion called Absorption. There was also the permanent fusion of Supreme Kai and his servant Kibito by using The Potara Earrings. Supreme Kai's personality was the one that stood out after that. For Dragon Ball Super we also see Kale and Caulifla fuse into Kefla who actually only seems to inherit Caulifla's personality.
combine both their attributes, from strength and speed to reflexes, intelligence and wisdom. When properly fused, the single being created has an astounding level of power, far beyond what either fusees would have had individually.
So, none of the fusees control the mind after the normal fusion. The being created after is a combination of both of their individual attributes and as such can think for itself.
Fusion dance = an entirely new being. And as far as the earrings go, it depends.
If one person is far more dominant in power and personality, then they remain who they are and are in complete control of the new form (old Kai and Kabito Kai), but if both fusee's are equal in power and dominance of personality, then similar to the dance, a new being completely will be created. You can tell if this occurs if you can hear both fusee's voices (Vegito).