Coding and testing schemes for binary hypothesis testing over noisy networks are proposed and their corresponding type-II error exponents are derived. When communication is over a discrete memoryless channel (DMC), our scheme combines Shimokawa-Han-Amari's hypothesis testing scheme with Borade's unequal error protection (UEP) for channel coding. A separate source channel coding architecture is employed. The resulting exponent is optimal for the newly introduced class of \emph{generalized testing against conditional independence}. When communication is over a MAC or a BC, our scheme combines hybrid coding with UEP. The resulting error exponent over the MAC is optimal in the case of generalized testing against conditional independence with independent observations at the two sensors, when the MAC decomposes into two individual DMCs. In this case, separate source-channel coding is sufficient; this same conclusion holds also under arbitrarily correlated sensor observations when testing is against independence. For the BC, the error exponents region of hybrid coding with UEP exhibits a tradeoff between the exponents attained at the two decision centers. When both receivers aim at maximizing the error exponents under different hypotheses and the marginal distributions of the sensors' observations are different under these hypotheses, then this tradeoff can be mitigated with the following strategy. The sensor makes a tentative guess on the hypothesis, submits this guess, and applies our coding and testing scheme for the DMC only for the decision center that is not interested in maximizing the exponent under the guessed hypothesis.