I get asked this question a lot concerning lateral (parallel to the action) moves and here's my answer: it depends on what is easiest on the operator. Generally, always keep the dolly chassis on the (camera) left. Why? Because that's where the eyepiece is. Keep the camera operator on the dolly as much as possible. An operator walking is a pain in the ass for him and you both. He unconsciously surges, you're trying to drag him down the track and watch the action and him at the same time to make sure he's safe. It's unneeded drama. Just keep him on the dolly. Whether you are pushing or pulling generally doesn't matter in your operation of the move, unless it's a particularly fast move, etc, so don't worry about yourself, worry about your operator. Personally, I rarely push from behind the dolly anyway. I push from the side so I can get my eyes as close to camera as possible in order to have the same perspective. Take note of what the shot is and where the camera is pointed for the majority of the shot and plan accordingly. If you need to, sit on the seat or stand on the sideboard, look through the eyepiece and do a test operation yourself. If you can do it, your operator can do it. This is one of the cardinal rules of pushing dolly-Keep the dolly under the operator.
Now, having said that, there are some lock- off situations where you can't put the dolly on the left. I'm talking about when an off camera actor is on or entering from camera left. This is one of those situations when your knowledge of eyelines comes into play. If you do a closeup and you know from previous shots that the off -camera actor is camera left, you should put the chassis on the right (or straight behind the lens) to give him room to do his thing. It's all about being aware of what's going on shotwise. You have to think like a camera operator. Learn as much as you can about eyelines, cutting, what's going to be used and what isn't, and blocking and pay attention. Then, you'll often know where the camera's going before anyone else does.