But does getting agreement from me, or anyone else for that matter, really mean very much?
Maybe pick some system and establish it for yourself? There are systems that'll make life pretty easy.
In one that I have in mind you could reason informally as follows:
Necessity: (b) is a necessary condition for (a)
Take (a) as premiss. Pick an arbitrary name, say c, drop the existential quantifier and assume p(c) or q(c).
The 'or' (now as a main connective) should suggest "proof by cases". So derive (b) from p(c).
The second case (deriving (b) from q(c)) follows immediately from the first case by simply changing what needs to be changed. Except for tying up some loose ends, you're done (at least in one direction).
Sufficiency: (b) is a sufficient condition for (a)
Take (b) as premiss. I see another 'or' (and it's already the principal connective). Now what?