First, consider this: Interior doors comprise a far larger expanse of shared-wall than you might think at first. If the wall you share with the noisy area is 80 square feet (i.e., the typical 8 feet tall, and then another 10 feet wide), then the door represents close to 20% of that area. Follow so far?
So what's the one weak point here? Interior walls have two sheet of drywall, one on each side, and the sound-insulating space between those sheets. Not great sound-proofing, but not the worst, either. The weak point: that hollow-core door of yours. Hollow-core doors have just two advantages. They are relatively inexpensive, and they are easy to swing. But hollow core doors punch through at the slightest impact. They feel and look cheap, and I don't mean inexpensive. When it comes to sound-proofing, one of the most recommended practices by residential sound-proofing experts:
Replace hollow core doors with solid doors!As long as you stick to the same dimensions, it is strictly a one-for-one replacement process. Screw and hinge placement should be the same, though not necessarily. I would make sure that the hinges matched up the same. And the screws need to either match up, too, or be far enough away from existing holes so that you can drill new, strong holes. If the new holes are close enough to the old holes that they merge, then you've got trouble. There is redundancy in hinge screw placement, so you could--if you had to--sacrifice a screw. But because you're dealing with the increased weight of a solid wood door, you really need all the strength you can get.
You will absolutely notice the difference in noise between a hollow core door and a solid wood door, believe me.