As one of the classic models that describe the belief dynamics over social networks, a non-Bayesian social learning model assumes that members in the network possess accurate signal knowledge through the process of Bayesian inference. In order to make the non-Bayesian social learning model more applicable to human and animal societies, this paper extended this model by assuming the existence of private signal structure bias. Each social member in each time step uses an imperfect signal knowledge to form its Bayesian part belief and then incorporates its neighbors' beliefs into this Bayesian part belief to form a new belief report. First, we investigated the intrinsic learning ability of an isolated agent and deduced the conditions that the signal structure needs to satisfy for this isolated agent to make an eventually correct decision. According to these conditions, agents' signal structures were further divided into three different types, "conservative," "radical," and "negative." Then, we switched the context from isolated agents to a connected network; our propositions and simulations show that the conservative agents are the dominant force for the social network to learn the real state, while the other two types might prevent the network from successful learning. Although fragilities do exist in non-Bayesian social learning mechanism, "be more conservative" and "avoid overconfidence" could be effective strategies for each agent in the real social networks to collectively improve social learning processes and results.