I don't want the dog desensitized to it, no.
Why not? This seems a little silly to me. I would like my dog desensitized to the stick, in that they still recognize it as an act of threat by the helper (although truly I think a good helper offers a good deal more threat through body language than actual stick threat), but that it's no longer any big deal. Although maybe it's a question of semantics regarding the word "desensitization". To me desensitization is a process that makes an animal less sensitive to a stimulus. We do it all the time in socialization. We desensitize our dogs to people, places, and things so that those things are not new or startling. Some dogs are naturally not sensitive to the world around them, and some are.
To my mind the stick is used in increments, pushing to a dog's threshold that I understand is felt and seen by the helper through the bite on the sleeve and other things the dog does. In working our dogs I've seen it used in a number of ways. Hitting a tight lead behind the head. I've seen helpers use hands. Pet the dog with the stick. Start with hits on the side of the dog. And following the stick hit/threat there is some kind of release of the pressure. The release of pressure can come with the release of the sleeve, or a change in the helper. I have one dog that when he would counter in on a stick hit the helper would collapse a little to the ground, so the dog could feel like he was dragging him down and winning their fight. I also feel it's important that if a dog should start to lose his grip during a stick threat, it's important to allow the dog an opportunity to counter before releasing the sleeve. The dog should win for countering or for not being phased in the first place.
And of course different dogs show different levels of sensitivity. One of my dogs you could beat all day in the head and he wouldn't even blink. There was no training for the stick. He just took it right away. I have another who is a good deal sharper. Training the stick was more important there.
One thing we did as handlers was to smack our puppies around. We hit them with hands on the sides, with the ends of their leashes, with sticks on walks, and while playing and turned a good smack into a sort of "Atta Boy". This way objects waving by their heads, physical contact to the body doesn't phase them. But of course these are just elements- none of which should involve threat when done by the handler. But being used to these elements can help the dogs to better focus when presented with a threat involving them. So rather than my young dog in protection having to deal with being a little unnerved by these elements at once- level of threat given by the presence of the helper, the noise, the movement, and the physical contact, all he really needs to focus on is the variable I want him to focus on...the level of threat presented by the helper. Usually when you see a dog being worked with on the stick, you usually see it being broken down into those elements. The movement (swinging the stick but not hitting), The noise (hitting the lead), the physical contact (hitting the dog in a less threatening way on the side), and then only finally on actually coping with the threat.
A good helper should be giving when presenting any kind of threat to the dog. The dog should always win the contest, regardless of how you are training. And no, I would not expect a puppy to be competing with the helper at this point in a fight. Of course any fight behaviors should be rewarded by your helper, but that's not really something most handlers can direct..because your arm is not in the sleeve and you cannot feel if there is compression of loosening. You can't really see if you dog averts his eyes to avoid or if he's staring right in the eyes of the helper.
So you have to communicate with your helper. There should be a purpose behind why he does what he does. He should be able to tell you why he slipped the sleeve, or why he decided to take the dog into a drive with the stick. Even if it's only just to see what would happen. One time will not ruin a good dog and it will provide information to your helper on what to do to adjust next time. A good deal of training is trial and error.