It's important to understand that advanced providers (paramedics, RNs, etc) cannot perform advanced interventions while off duty. Things like IVs, needle decompression, etc require medical direction, which we don't have while off duty. That is a major reason why most people don't carry that stuff off duty.
I carry a small first aid kit. It had a tourniquet, gauze, some saline for cleaning, etc. I also have some OTC medication for personal or family use. Things like aspiring, benadryl, ibuprofen, etc.
I do keep my work gear bag in my truck which also carries some personal preference ALS equipment, including a ARS needle. My medical director has chosen to extend our medical direction to us off duty. This works by having our dispatch remotely clock us in if we happen across a serious emergency while off duty within our county. This is extremely helpful as many of our employees are also volunteer firefighters. If one of them responds (or happens to be nearby off duty) we can have them clocked in remotely and they become fully covered employees.
I've used this more than once on a dual medic truck to bring on an off duty EMT to drive so my medic partner and I can both be in the back