There should be a white wire (ground) and a blue wire (voltage) going to your brakes. The white wire hooks to a ground bus in the trailer that is connected to the ground wire in the truck connector.
No, you do not need a trailer battery for the brakes to operate with the exception of the breakaway switch.
John W. Irwin
2005 Classic 28 "Sabre-Dog III"
2013 Silverado 2500HD Duramax/Allison LTZ
WBCCI #9632, TAC TX-10