I usually let the guys dumpster dive for metal. I don't give them permission outright, but I do avoid blindly throwing stuff over the rim if I see their truck still nearby. Most often they come after hours anyway. Personally, I prefer to see the material taken for recycling by those guys than hauled to a landfill. The guys scrapping metal typically realize that they're trespassing and work quick so as to not get caught. As long as you're not intentionally making the dumpster unnecessarily dangerous or alluring, liability is pretty low.
Even if you catch a dumper's license plate, you're not likely going to get the police to care and it'll cost you more in time lost to try to litigate. Other than that, your best bet is to shorten the amount of time the dumpster is around to a week or less. If I'm doing a full construction trashout, I'll easily fill a 30yd dumpster in a day or two. I did a whole house remodel (full kitchen, new paint, new hardwood, scraped popcorn ceilings throughout) where I staged all the trash in their vacant garage during the month. When complete, I ordered the dumpster, filled it, and had it gone the next day. I once had 3 laborers fill a small dumpster in 45 minutes while it was still connected to the waiting truck.
It probably cost an extra $50 in labor to move the trash twice. A bag or two extra from a neighbor adds nothing overall, so I don't sweat it. Heck, I've done it myself. But if your framer is dropping truckloads of his own trash in your dumpster, you don't have a dumpster problem--you have a people problem.