e.g., each node can generate its own subset to check, so you only need to distribute the program and input. That can be done using a multicast tree of some kind which brings distribution costs back down by a log.
If your node population increases exponentially with input size, you're still running into the speed-of-light delay problem mentioned above (cube root of exponential is still exponential). If your solution is polynomial time with polynomially many nodes, your space cost is also polynomial.