So a more literal translation of 左側を歩きます would be 'run through the park', while 左側で歩きます would be 'run at the park'. I understand particles don't really have perfect translations, but do you think it is useful to think of these sentences in this way?
Also, let's say for example, someone is walking along the left side, you could say その人は左側を歩きている, but if someone were to ask at that moment, "what side is he walking on?", you could say 左側で歩きている. Is that correct?