1. The two sides are too hostile to one another, and disregard everything the other side says, while attacking the character of those who hold opposing views.
Because of this, both sides learn little from one another, and can get stuck in narrow and often sloppy thinking.
2. The debate is not held publicly.
Because authorities are scared of vaccine skepticism they are unable to rationally deal with it in an intelligent, open way.
And so they pretend there are no doubts over vaccines except among rabid, irrational "anti-vaxxers" whose views are said to be unscientific and way off-base.
As a result of this attitude and the hostility directed towards vaccine skeptics, the skeptics themselves project the same kind of loathing back.
Open and frank discussion would help heal this rift and would also provide the public with both sides of the story which would also help restore faith in the institutions which are currently seen as unresponsive to public concern over the issue.
A full and frank debate would shed light on the matter, revealing where more knowledge is needed, rather than the current situation where one side says all is well, no need to change and the other says the whole thing is a disaster!