I am writing some random thoughts. Your experimental report is perplexing.
I just noticed that in your original post you mentioned a semi permeable membrane. I do not recollect ever seeing a semi permeable membrane during electrolysis of brine. Of course, I may have had a sheltered life.
You picture has 3 colors which I assume are not there actually, but represent your theory as to the location of molecules.
Dredging from past memories, it seems to me that chlorine gas smells like bleach.
http://en.wikipedia.org/wiki/Sodium_hypochlorite Based on the previous link, as long as you have the cathode and anode separated, sodium hypochlorite should not form. The other compound sodium chlorate should not form if you keep the electrodes far enough apart as well.
http://en.wikipedia.org/wiki/Sodium_chlorate Again from my memory, the formula for electrolysis of water is as follows.
2H2O = 2H2 + O2
Therefore we get a column of hydrogen twice the size of oxygen.
And again from my memory, the formula for the electrolysis of brine is as follows.
2H2O +2NaCl = H2 +Cl2 + 2NaOH
That produces a column of hydrogen the same size as the column of chlorine.
Remember that according to the link for sodium hypochlorite the following reaction should not occur if the electrodes are far enough apart.
2NaOH + Cl2 = NaCl + NaOCl + H2O
But, I got to believe that some of the hydrogen and some of the chlorine would dissolve in water. The amount is probably small, but it will still give you the smell of bleach. I would think it would eventually dissipate. Maybe if you keep the solution warm but not boiling and the electrodes far apart, then the gases will not stay in solution very well. I still believe that when you use up all of the sodium chloride in the solution and start having oxygen given off because of the electrolysis of the remaining water, that most if not all of the chlorine that was available will be released as a gas.
As always I add the disclaimer. I have not done this in the laboratory and would have to defer to those who actually had some hands on experience. Theorists most always do a lot of hand waving and unfortunately in this case I am not a practitioner. Maybe someone else will weigh in with a link or experience that will be useful.
By the way, I was hoping for the electrolysis of pure water and pure salt to make pure sodium hydroxide would work. One can not always trust the purity of stuff purchased on EBAY. How quickly hopes are dashed.