If the public became aware of GMOs as they are now demand for non-GMO food will increase. This means that companies that don't use GMOs will begin to label that their food does not contain any GMOs in order to increase their profits. Other companies will be forced to follow along and label their food as non-GMO in order to not lose market share. If you look at this through a game theoretical approach it is easy to see that the equilibrium will be to label their food. Hence, we will know which food contains GMOs by a non-label.
One response I always get from this is how do you know the company won't lie? Well, for one thing, fraud is illegal. No one is advocating fraud. In reality, what would happen is that with the fear of fraud there would be a demand for a company to test these foods for GMOs, and without the approval of the company the label of non-GMO means nothing. Now, we have companies labeling that their food does not contain GMOs and a third party company approving which food can be labeled.
Another thing I hear is that this way of the free market handling these food labels will hurt mom 'n' pop stores. I don't see how this follows at all. They claim that they won't be able to afford the labeling. However, why would a mom 'n' pop store need the labeling at all? Usually these small mom 'n' pop stores are in business due to a loyal customer following where the owners know the customers. If they are selling their own food I think a simple sign above their door about non-GMOs would suffice to convince loyal customers, and if they are selling others foods they don't have to pay for the labeling at all. Furthermore, the way the free market handles this doesn't force them to label anything if by chance their food does contain GMOs.
Moral of the story is whenever someone is arguing for more government regulation try first to think how the market would handle it. Usually it is not overly-complicated and almost common sensical. If it is not, you might be over thinking it.