Yeah, they call it "passive". It looks like it's basically like unflagging for PvP in WoW.
I was reading a Q & A thing, and someone asked "So if another person kills me, does it cost like $5,000 like it does in single player?" (it actually takes a percentage and someone clearly doesn't dump their cash in the market to prevent this from happening, but that's neither here nor there)
And, no. No it doesn't.
When someone kills you in GTA:O, you drop all of the cash you have on you. But not what you have in the bank.
You can then put a bounty on their head, if it pleases you to do so.