How powerful are nondeterministic constant-depth circuits?
A polynomial amount of nondeterministic bits is enough to encode the computation of a nondeterministic polynomial time algorithm. The only thing we need is to check if a given string is an accepting computation which is syntactic task that can be performed by a polynomial-size $\mathsf{AC^0}$ circuit (in fact a polynomial size CNF can do this).

Another way to look at this is to consider the Tseitin translation from arbitrary circuits to CNFs which has a polynomial size increase and uses only polynomially many new propositional variables. 

If you look at the $\mathsf{NP}$-compeleteness proof of CNF-SAT (or 3SAT) you see that the part that checks if a given CNF is satisfied by a given truth assignment can be computed by an $\mathsf{AC^0}$ circuit (IIRC $\mathsf{AC^0_d}$ circuits can be evaluated by an $\mathsf{AC^0_{d+1}}$ circuit).