Dispatcher,
the policy we use in my airline is to let in as many luggage come, unless they are too heavy to be lifted to the overhead bins or too big to fit. When there is no more room available, we start sending them to the hold. Usually no more than 2-3 per flight, and if it's full. But you'll admit it is not fair to let in people with more than the allowed weight/pieces and then deny the last to board who maybe just has one piece of the regular weight.
Would be much better to screen before the obvious irregularities like people with 3 pieces or enormous luggage (especially when the flight is full, as it often is the case).
When we send something to the hold we don't need to let ground staff know, because they usually are there and see what happens. And we usually don't complain to them, just let them know what happened if the situation was very bad. As I said, it's no problem to us: baggage get on board until they fit. After, they go to the hold. End of problem. Pax usually don't complain because 1) they can see there is no more room available 2) they can see we do all that is possible to help them.