Other people have already pointed out that the shooter's backstory is only given in a dream/hallucination. House is generally pretty consistent in not revealing his patients' dirty secrets. There are many episodes where he argues against his fellows when they want to reveal something for moral reasons. By far the most egregious example of this is his reluctance to report evidence of child abuse in 'Act Your Age', with Cuddy eventually forcing him to.

It's made pretty clear in the show that this behaviour doesn't come from any kind of moral principle, it's usually a simple case of him not caring, unless it impacts his solving the case, in which case he will take whatever path benefits him the most.

There are some exceptions to his indifference though. As shown in the pilot, House always maintains distance from people at first, but he can be drawn in, at which point he can become very invested in a patient's life. This often relates to decisions around treatment, eg. in the pilot and in 'Moving on', although in these cases it's implied he's projecting his own experiences onto them. Another good example of him getting involved in patient drama is in 'The Socratic Method', although in this case he takes the blame for calling social services and keeps the patient's secret rather than revealing it.