We study the power of \textit{local information algorithms} for optimization problems on social networks. We focus on sequential algorithms for which the network topology is initially unknown and is revealed only within a local neighborhood of vertices that have been irrevocably added to the output set. The distinguishing feature of this setting is that locality is necessitated by constraints on the network information visible to the algorithm, rather than being desirable for reasons of efficiency or parallelizability. In this sense, changes to the level of network visibility can have a significant impact on algorithm design. We study a range of problems under this model of algorithms with local information. We first consider the case in which the underlying graph is a preferential attachment network. We show that one can find the node of maximum degree in the network in a polylogarithmic number of steps, using an opportunistic algorithm that repeatedly queries the visible node of maximum degree. This addresses an open question of Bollob{á}s and Riordan. In contrast, local information algorithms require a linear number of queries to solve the problem on arbitrary networks. Motivated by problems faced by recruiters in online networks, we also consider network coverage problems such as finding a minimum dominating set. For this optimization problem we show that, if each node added to the output set reveals sufficient information about the set's neighborhood, then it is possible to design randomized algorithms for general networks that nearly match the best approximations possible even with full access to the graph structure. We show that this level of visibility is necessary. We conclude that a network provider's decision of how much structure to make visible to its users can have a significant effect on a user's ability to interact strategically with the network.