To BLL’s point…if you have the LLC and an insurance policy covering the LLC (building) and you as an individual act as a property manager, and are negligent, you are creating an additional exposure to loss . However, some insurance policies have verbage that picks up the manager of an LLC as an additional insured. If this is the case and your loss is covered and not excluded via your policy then your personal exposure you created should be covered…make sure you have adequate limits…$1-5mm. If however, your loss is excluded under your policy then you’re in for a suit against your personal assets…as is the LLC. An example may be lead poisoning or mold bodily injury.
There will always be a creative lawyer if something/someone gets hurt. The best method of managing your risk is to be proactive and control the hazards before a claim.