if by "them" you mean dogs then they either trust you enough to do stuff with their privates while they're on their back or they don't and that's that... on the back is an uncomfortable position for most dogs in both physiological (do make sure to not apply any force on their pelvis in this position) and mental reasons, it's a very vulnerable position to be in. also, you don't "ride" a dog.. it's just a tie in a different position. riding (as in sliding up and down the shaft) does nothing pleasure-wise to them, it can hurt them tho.
all in all, doggy-style is preferred to most dogs. i'm not claiming there aren't any exceptions or outliers, but it's named "doggy-style" for a reason.