Another sign that they have swallowed is when they lick their noses. Sometimes, I will pry open Spot's mouth after I think he's swallowed, and then release, and he will swallow for sure. Depending on the kind of pill, it might be bitter. If so, you may want to try pill pockets or empty gel capsules (some pharmacies and most health stores carry them) so that the kitty doesn't taste the medicine. On my stubborn, squirmy cat, I try to pin her between my legs, pry open her mouth, and push the capsule as far back as possible without her gagging. I close her mouth *quickly* and hold it gently but firmly closed until her tongue licks her nose. If I don't hold her mouth closed, she will start a chewing response and the whole pill goes up in foam and I end up with half a cup of drool on my lap.
The other reason the vet doesn't have an issue is because your cat doesn't trust him or her enough to fight. With you, she knows that you won't do anything to hurt her even if she is uncooperative. She doesn't have that same faith and trust in the vet.
Oh yeah, I also feed or give my cats treats after the pills so they have something to replace the taste and texture of the pills. Plus, I stay on their good side when I give them Pounce.