All of the trailer running and stop lights are powered by your truck battery when hooked up to the trailer cable.
The trailer battery is REQUIRED by DOT for trailers equipped with electric brakes for safety requirements. Like said above if your run-away trailer causes damage to property or worse then your Insurance may have a field day with your claim if it is known you DO NOT have the required battery on-board required by DOT to power-up your electric brakes if you have a disconnect going down the road.. Might get expensive for you to settle a lawsuit on your own. Too big a risk for this ole cowboy...
I would not pull my trailer with electric brakes anywhere without a working trailer battery setup installed.
BTW - Not all trailer are setup the same I guess.. My emergency brake pull switch assy on my two 2004 and 2008 trailers provides 12VDC from my on-board trailler battery to all of my trailer wheel mounted electric brakes using the same wiring the truck mounted brake controller is using.. If left "ON" it will discharge my battery pretty quick and probably do harm to the electric brake module as each electric brake assy pulls 3AMPS of current when 12VDC is applied. My fifth wheel trailer with four brake modules will pull 12AMPS and my OFF-ROAD trailer with two brake modules will pull 6AMPS when 12VDC is applied.
Roy Ken