Here, private security guards have (as Brian pointed out) the same authority.. no more.. than private citizens. HOWEVER, it is very annoying to be shot or pepper-sprayed by someone with no authority to do so.

To the point of this thread: IMHO "authority" and thence control of the situation comes from the individual's bearing. His tone of voice, voice amplitude, body language, etc. Perhaps augmented by "symbols" of authority such as weapons, uniforms, possession of the survival kit, etc.

There are verbal/vocal techniques which are often thought to enhance this "authority" and get better compliance:
Don't shout unless absolutely necessary. Talk slower, and in a deeper tone if you can.

Don't instruct in a "generic" manner. Not "someone get a blanket" but rather "YOU sir/mam ... (eye contact, even a light touch on the arm/shoulder) get a blanket and bring it over here please, NOW please"

If time allows, start with the "foot in the door" technique. Compliance snowballs. Start with simple instructions with a high chance of cooperation - "Step back 3-4 paces please".. "Gather over here a moment please".. "Sir, hand me that chair please", etc. In the "panic" type situations, once simple compliance starts there is generally a reluctance (perhaps even a physiological impediment, we think) for the panic-stricken to SHIFT. If you can establish authority/leadership initially and on no-brainer tasks, folks are predisposed to remain with you rather than SHIFTING to other leadership (including shifting to that internal voice of panic).