They often do work without batteries, they are esentialy low quality plastic versions of the old school Victrola record players that rely on the record needle directly vibrating a diaphragm to make sound and thus need no power just somthing to spin the record which could be a pull string in a doll like this.

The same concept is used on those old animal sound spining toys, except those had multiple tracks arranged in concentric spirals on the record. When you pointed the arrow at the animal whose sound you wanted to hear you were actualy lining the record up so the needle would fall into the groove for the right track.