Actually, I have heard that it does make a difference on what side you play the reed on. Reeds should have a thin and a thick side. This is what improves the overall sound quality, along with having a noticable spine and the "windows" on both sides. You would want to play on the thin side facing down. My instructor who makes reeds always crosses the thread of the reed on the thin side. This allows me to distinguish what side is the thin and what side is the thick. However, I'm not sure if ALL reeds work this way, especially machine made ones. But one of the fellow oboists purchased a machine reed one time, and got a pretty harsh sound out of the oboe. She asked me if I could figure out the problem. Sure enough, I suggested that it may help if she were to turn the reed over, as she was playing on the side where the thread did not cross. After making her adjustment, her tone was much better.
It's a rule that I have always tried to become aware of. But of course, with every rule, there are exceptions. I have played on reeds where both sides have no difference in quality or pitch. To really get the full potential out of your reed, I would purchase from a private reed instructor, that way you can tell him or her your individual needs, not just soft or hard.
I would purchase a reed knife and a plaque in order to shape your reeds as needed. Purchase "The oboe reed book" by Jay Light. This book will teach you how to make reeds. If you don't want to learn how to make reeds it will help you correct your store bought reeds so that you can get better tone quality and pitch out of them. Some people play better on hard stiff reeds and others on soft reeds. I'd try a variety of reeds and find which works better for you. As far as a thicker side working better I think that's a matter of opinion.
The opposite sides of the reed, should theoretically be identical, so playing on either would make no difference to the sound. However, the cane reeds are made from are natural products, with natural deviations in density/flexibility across the whole reed, so rit can be hard to make the two sides identical.
In my experience, one side of the reeds tends to scrape much more freely than the other.
So to answer your question, no, on a well-made reed, it should not matter which side you play on, but if one side is better than the other, so be it!