Zero-knowledge identification of the majority commitment
Here is one difficultly such a protocol can have: say we do have a joint computation protocol that takes $N+1$ parties (Alice and her $n$ people); at the end of the protocol, Alice gets a list of the parties in the majority without learning their secret.

Here is how Alice can attempt to cheat: extend the protocol to $N+3$ parties, and have the extra two parties be two artifical votes controlled by Alice; thing0 will submit a 0 vote, and thing1 will submit a 1 vote; other than that, the protocol continues as is.  thing0 and thing1's vote will not change which bit was in the majority; however, when Alice gets a list of the majority party; either thing0 or thing1 will be in the list; by that, Alice will learn the secret bit.

Any such protocol will need to avoid this somehow; either by simply not being extensible to N+3 parties, or having each party get a count of how many people are participating (so that they'll know that there are two extra ringers)