Swimming with chest cold ?

If it's that deep it is best to avoid the swim anyway.  I feel your pain though and have taken a swimmers towel/chamois with me to cough/sneeze into at the wall.

spit it into the gutter.  the filtration system will take care of it.

yuck...I hope people in my pool aren't spitting their fleghm into the water...I am sure they are but I try to not think about it!
General rule of thumb...if you feel sick above the neck (sore throat, stuffy nose, congestion etc...) it is usually ok to workout, but cautiously.
Anything below the neck (chest congestion, coughing, body aches etc...) it is best to rest.
I can do anything but swim when I get sick, but that is a definite no. If I can barely breathe in air, I know for sure that I am not doing myself any favors being in heavily chlorinated water (not like any of us swim in Fiji water) so I stay away.
I once didn't swim for almost a month due to a nasty cold that got worse before it got better. No way could I have swam during that time.
If you are feeling that sick...why not take a few days off, not stress your body anymore than it already is (keep your germs away from others) and come back quicker and not risk a relapse.
Feel better soon!