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In my personal experience NO Staffie's do not drool after long walks over the years i have owned 6 staffies and none of them drooled.
The short answer is no. Drooling is a reflexive reaction, not a behavior. For example, dogs (and humans) start to salivate in anticipation of food. Drooling, or excessive salivation, can occur for a number of reasons besides food, such as a physical reason, heat-related drooling, and stress. Lets start by investigating the least likely reasons for Mollys drooling.