I carry a personal bag and a box of gloves in my trunk. I use the gloves for all kinds of things from checking oil to cleaning dishes.
Personally I carry my bag because I HATE the way the issued bags are laid out, I hate that if I use even the BP cuff I have to retag/inventory the entire bag, and most of all I hate the equipment we use, IE the cheapest stuff you can get.
The biggest reason I can give though for carrying a bag is that I'm OCD about organization and my supplies.
^ This. First, the bags at work are crap. Half are falling apart, and they're small bags with a large amount of crap shoved into them. Also, they lack a couple things I feel they should have.
At my volly company, we don't have a BLS house bag. Yes, that's right, no bag. Yes, we had a crew recently walk in with only a stretcher and a bag with a BP cuff/steth/a few 4x4s in it, and it was a cardiac arrest, and it was a mad scramble to go back and get everything. I try to avoid that embarrassment.
So, my bag does carry things like Adult/Ped BVMs, airways, etc.. Things for bleeding control, and yes, Adult/Ped collars. I also carry a couple things that no one else seems important enough to carry, like a thermometer and a ring cutter. Personally, I feel the most important part of my kit is the emesis bag, because I hate having to clean up puke. Ironically, they always seem to miss the bag. <_<
However, I can say that I can probably count on both hands the number of times I've actually used it in my personal life, and never more than a puke bag or a 4x4 and some tape.
I'm actually allowed, per my volly company, to carry ILS equipment. IV supplies and King tubes, but no ETT, due to not having capnography handy. I just see no need for any of that. Outside of work/volly, I follow Veneficus' policy.