> How long would it take for one of the servers to see every piece of the password, on average?

Strong secret sharing schemes are secure against this kind of attack. The schemes are designed so that you cannot infer any information about the original secret from any of the pieces. That holds even if you get a single piece from each of multiple independent "splits" of the same secret.

Some examples of such schemes are on Wikipedia and they fall into the "simple but clever" category: http://en.wikipedia.org/wiki/Secret_sharing#Trivial_secret_s...