Some companies that offer odor-free bully sticks want to suggest that they have somehow unlocked a proprietary magical process to make their bully sticks the least stinky on the market. But other companies actually do share a common thread in their explanation: thorough drying.
According to Best Bully Sticks, the difference between their odor-free bully sticks and the regular ones they also sell are how long they are hung to dry. "The difference in how they are made is that the odor-free bully sticks are dried longer, lessening the moisture in them."
Bow Wow Labs similarly notes, "The odor of the sticks is related to the amount of moisture in them."
An article on Rover echoes this idea, with a vet quoted as saying, "The longer they are dried or the more moisture removed by cooking, the less odor they will have."
Personally, I've only ever given my dog odor-free bully sticks. Given that the difference is in drying technique, it feels like a pretty safe (and less noxious!) choice to me.