Two significant hazards IMO
Hydrogen and its relatives, like H2S (Hydrogen sulfide). Hydrogen atoms rise fast and aggressively seek any escape path they find. H2S is denser but the gas is only an element of concern during catastrophic overcharging.
LEAD and its various components. Venting has a pb carryover factor. Usually the lead coalesces and precipitates fairly close to the vent. But the entire inside and outside of the containment vessel should be treated as a very real pb hazardous waste site.
READ THIS CAREFULLY por favor. There are two types of lead. Pure lead IS NOT NEARLY as poisonous as ORGANIC LEAD which is the type of lead found in lead acid batteries. The very worse form of organic lead commonly found was tetra-ethyl lead which one single droplet on exposed skin caused death.
So, in my opinion caution focuses on proper ventilation of flooded lead acid batteries and not on the dangers of handling organic lead. Rub a cloth inside a battery storage box. See the black smudge on the cloth? That's organic lead in sulfate form and when it gets on your skin it is absorbed.
Both ventilation AND skin protection is important. I use nitrile gloves and a painter's disposable mask.