Lisa:
The most accurate info (IMO) that I found on this topic was that teething wears down the immune system and this wearing down can cause many effects different for each body and each situation.
The body sees these teeth as new and foreign and goes into full-gear protection (code red) against foreign objects. Because the body thinks teeth are foreign (just like the body does for an allergin), the body may try to "attacks them" like it "attacks" allergins and this "attack" causes allergy-like symptoms (runny nose, congestion, cough, etc)! These symptoms are how the body feels it may be able to get rid of this "bad" foreign object: fever (shows the body's deffenses are working over-time, ie: Over-heating, to get "the bad" out), congestion (mucos is the body's super-highways to try to move "bad" things out of the body), sneeze/cough (expell the "bad"), etc.
Over time, all of this over-time work of trying to expel that "bad" from the body exhausts the immune system's resources making the teething child more vulnerable to getting virus/infection. Thus, they are more likely to get sick during teething.
So: CAN YOU GET SICK FROM THEM (ie are they contagious)? If their immune system can't fight off a virus (ie: cold) because it has been over-worked and tierd, and then virus takes over, THEN YES, THEY ARE CONTAGIOUS. BUT, if it is just "allergy" symptoms, there is nothing there to catch. The problem arrises not only in knowing what exactly is going on (why many drs. say teething doesn't make babies sick) but also in what possibly can develop: cause and effect. "Allergy" symptoms can harbor germs (thick mucos and congestion make prime breading ground for bacteria to grow in ears, throat, sinuses; cough/sneeze can irritate throat, lung and nose lining breading its defense against bacteria, etc) that then can turn into a virus (which in turn can be contagious)!
So: See teething as the baby's body going into battle! If the body is well ready for battle, it can defend away or push back the symptoms BUT if the body is not well prepared, it may loose a battle or two or the whole war!
I know this does not answer your questions on whether you will catch something from your teething baby or not, BUT it does show how it is possible.
As far as how long the symptoms will last, again, it all depends on how your baby's body can handle the stresses!
Confussed? This is why drs can't even give a good answer. However, my pedi says (and most of this explanation is from him with added research by my psty) THAT MOST CERTAINLY TEETHING CAN CAUSE SICKNESS BUT DOES IT ALWAYS OR EVEN OFTEN, THAT DEPENDS ON THE CHILD AND PARTICULAR CASE!
Good luck!
T.