Anna, 
As for your question regarding "how a child who is not vaccinated and gets measles can spread the disease to the vaccinated children? They should be protected, right?"  I'm sure you're already well aware of the answer as you're so well read in the field right, but I'll answer all the same. For one, the measles vaccination, as an example, is not a 100% effective, nor does it claim to be. So, sometimes a vaccinated child can still get the measles. Also, due to a variety of health reasons, some children cannot get vaccinated and must rely on herd immunity to protect them. That protection is diminished when fewer people are vaccinated putting them at greater risk too.  No conspiracy here, just facts. Please don't ignore them, heed them for the good of our children.