There are two distinct definitions, or methods of trading dogs.
The adjective "trade dogs" comes from the early to mid high school years, where young children used code to define when they would like to swing their fists. It was often used as a threat to extort lunch money or prey on poor defenseless nerds, while slamming them upon the hard metal surface of a locker.
At this point in time, this term is used as a greeting, completely meaningless inside joke or meme between friends. In this respect, to trade dogs can mean anything from joining a raiding party in World of Warcraft, to hanging out and smoking weed. It is absolutely imperative to use CAPS when trading dogs in this manner.
Bully: Hey punk.. wanna trade dogs?
Nerd: HAHAH what?
Bully: *swings at nerd*
Tornizzle: YO MANG WANNA TRADE DOGZZZ?
Dwingo: YA WHAT YOU GOT
Tornizzle: I GOT A GOLDEN RETRIEVER
Dwingo: Okay um, where are we raiding dude?