@K-ballo dug up this history: In N2997, the original wording was:

but they synchronize on the shared associated state

It seems that "associated state" was renamed to "shared state" later, and so the construction became "shared shared state".

Can we find a better word than "shared" for the first occurrence? Maybe "the common shared state"?

I think dropping the first "shared" should be fine, there is only (at most) one shared state involved.

Anthony Williams: "That looks like a simple editorial fix to me, especially since it's in a note."
Lawrence Crowl: "Agreed."