well I wouldnt... from what I understand the fumes that are coming out are pure chlorine... but I mean if you took the mms solution mixed with water and put it in a mister... then inhaled the mist, I dont think it would kill ya. I personally wouldn't do it. I mean people do the same with Hydrogen Peroxide.
Chlorine gas is potentially very damaging to the lungs. I would avoid breathing too much in. No need to be paranoid about it.....but it is best not to sit there and inhale the fumes.
Just as high concentrations of ozone will burn the lungs I would thing Chlorine Dioxide would also oxidize the lungs. It would be much wiser to use Colloidal Silver with a nebulizer.