It's not that there is any magic to having them on one side per se. (I personally don't want my dog crossing in front of me, so I want her to stay on the side that I put her). I think the advice that you've been given is great. My version goes something like this:
Leash tightens, you make a sound "Pssssh" then stop and wait. As soon as the leash loosens (most likely because they turn and look at you ("What the heck, why'd we stop?") causing the leash to slack, you continue walking. When the dog gives you want you want, i.e. takes a couple of steps on a loose leash, make sure she knows that's what you want. Praise, praise, praise. The key is to let the dog know when they are right and when they are wrong. Once they understand, you can say Psssh and they'll put slack in the lead without you stopping. The sound was always a precursor, so they'll know what's coming and slow down of their own accord. Otherwise, you're just stopping. If you're like me you do lots of actions that don't have meaning necessarily. You stop at crosswalks or turn or whatever lots of times. Those don't mean you are waiting for your dog to do something. So, its important for the stops to remind them to loosen the leash are different.
My main point though is about your own perspective. Its semantics, but I think its an important distinction. Teaching a dog an action is much clearer and easier to obtain than teaching the absence of an action. You will be a better trainer in all things if you decide what you do want, not what you don't want. If all you want is the dog not to pull (which is perfectly fine, you get to decide the parameters), then the positive action (not positive as in "good" or "bad," just as in "something" or "nothing") that you want is the dog to walk on a loose leash. It's hard to explain to a dog, "Don't do that. ...or that. Ugh, I don't like that either. Just stop being annoying." It's much easier to teach the dog an action. "Walk here." I personally like clicker training, which is nice and precise even if you aren't trying to get a competition heel, just a nice loose leash walk. However, German Shepherds are so stinkin' smart, you can usually get away with less than ideal training and they'll totally pick it up. Sometimes, I get complacent with my dog and then every once in a while when I don't get what I want, I realize, I'm not explaining myself very well and I go back to the beginning and try again.
Good luck. Let us know how it goes.