I'm not sure what the hospital might have to do, like report it to some agency

, but I understand wanting to know. My DD was in the ER a few months ago, and at one point we got a diagnosis, plans were made to admit her and treat her, and then before anything was done, it was discovered that they mixed her up with another patient. No harm done, the doctors apologized profusely, and we were relieved it wasn't HER diagnosis. However, we did wonder if it was reported to __________?? Some department in the hospital, or some agency, because IMHO they SHOULD be looking at why it happened, how it happened, and how to prevent it from happening again.
Another time I picked up a prescription at the Pharmacy, and days later when I needed it and opened the bottle, there were larger pills than I had had in the past. Turns out it was the correct medication but 4 times the dose. I noticed right away, but what if someone hadn't? (My mother was prescribed twice the dose I take, and had to stop because they made her light-headed and off balance...not good for an 87 year old woman with walking issues anyway.) Do I want compensation? Of course not...I didn't take them, and no harm done. But they did say that it would be "reported" (at least within the pharmacy co)...whether it was or not, I don't know.
The reason things should be reported is to prevent something potentially more harmful from happening. The next time, private information that gets into the wrong hands could do a lot more damage. So maybe the OP's particular situation didn't do any damage to their privacy, the error, IMHO, should need to be reported so that at the very least, the hospital is on alert to prevent it from happening again.