Do airline transport pilot trainees ever fly a passenger-less aircraft?
When pilots are trained for flying in an aircraft type they have not flown in before, they go through type rating, then line training. During the type rating, they learn where all the instruments and buttons are, what starts which system, how the plane behaves, how to land and auto-land etc.

During line training, they spend time in the cockpit together with an instructor pilot, usually a senior captain. Most airlines have a 3-person crew in the cockpit when a new F/O is receiving line training. This training is about the airline specific bit of flying, company procedures etc.

Note that before going to type rating, the pilot already has at least a commercial flying license: they can fly already. Depending on the country, the type rating can take place in a Level D simulator for 100%. Some countries still require a couple of actual flights in an aircraft, but there is an increasing trend to use Level D simulators only, for two reasons:


Back when the airlines paid and arranged for type ratings, it was relatively simple for them to arrange for an aircraft for a training flight, empty or not. But now a majority of type ratings is self funded, how does a student find the finance to charter an empty aircraft? 
The realism of Level D simulators is incredible. My office used to be not far from the simulators - every once in a while I would hear the siren for the sim coming down at an unscheduled time, for an unscheduled break: the stress level inside is real, you reckon that you are really 30,000 feet up with an engine on fire, alarms sounding, actual (non-toxic) smoke entering the cockpit, etc. After clearing the emergency, some students needed time to reflect.




Boeing predicts that we need 30,000 new pilots every year for the next 20 years. How do we streamline it such that all competent persons can become pilots, not only people with enough money to charter a plane?

As @user71659 points out, at airlines that only fly widebodies, standard entry position is a second officer/relief first officer/cruise pilot on a long haul plane like the 777. (Cathay Pacific, Singapore Airlines, Emirates, for example).