How to add a barrier with access to roll in and out trash bins in my 'public' driveway?
A few ideas, not knowing what the area really looks like. 


Over time seek to identify the various mystery bins and mark them with names or house numbers in permanent ink. When they appear on your property you can either bring it up with your neighbor or ask the local police for help. This may also work through peer pressure. If your neighbors see their names sitting on your property they may be less likely to violate your space. (I find that turning on my hazard flashers is the best way to deal with tailgaters--the unwanted attention very often mitigates the poor behavior.)
Fabricate a temporary gate, such as suspended cables with wire mesh between, that you can install the day before trash day each week and roll up when not needed. Put it up, take it down. Like when training a puppy, your neighbors will become conditioned to do something else.