If you properly collar condition your dog, there should be no difference if the collar is on or off. If your dog obeys when the collar is on, but not when it is off, he is colar wise, and you hosed the training.

That being said, I would not let the pup off check cord unless it was in a place where traffic was not a problem. Call the pup into you every now and then, give them a treat, then send them on their way.

Biggest mistake people make is only calling the dog in to leash them up. Think from the dog's point of view, why come in when he knows the fun is going to end and he is going to get leashed up. Dogs are not all that complicated, but you have to look at stuff from their perspective.