Communication allowed in the NC^i model? I want to verify if I am correct in my interpretation of the $\mathrm{NC}^i$ model.

In this model the input is given to at most $O(n^k)$ processors and each processor takes at most $O(\log^i n)$ time to arrive at a solution.

Can the processors communicate on any intermediate data?

How is the result whether accept or reject announced?



A fine grained problem

So the output of the algorithm can depend on partial results computed on previous processors? 

There are $O(n^k)$ processors. So even if each processor computes $1$ bit of intermediate data that is useful for all processors then reading the intermediate results will itself take $\Omega(n^k)$ time then there is no way processors can finish in $O(\log^in)$ time.

So what limitations are there in the length and dependency on intermediate results?
The model of parallel computing used in the definition of Nick's class is PRAM (see this wiki page, note: particular policy for handling read/write conflicts is irrelevant for Nick's class). In this model, the processors share memory and a common clock, but are otherwise independent. Therefore, any communication that happens between processors uses memory as the channel. The result of the computation can be announced in some pre specified register.

Edit:

Note there are a couple ambiguities in your statements: to show that a problem belongs to $\mathrm{NC}^i$ you can choose $k$ (the power in your polynomial number of processors) depending on the particular problem, while $i$ is fixed. Also, all the quantities should only be specified up to a constant, e.g. $O(\log^i n)$ time and $O(n^k)$ processors.