We study the power of local information algorithms for optimization problems on social and technological networks. We focus on sequential algorithms where 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. This framework models the behavior of an external agent that does not have direct access to the network data, such as a user interacting with an online social network We study a range of problems under this model of algorithms with local information. When 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 local algorithm that repeatedly queries the visible node of maximum degree. This addresses an open question of Bollobas and Riordan. Moreover, this result implies polylogarithmic approximations to problems such as finding the smallest subgraph that connects a subset of nodes, finding the highest-degree nodes, and finding a subgraph that maximizes vertex coverage per subgraph size. Motivated by problems faced by recruiters in online networks, we also consider network coverage problems on arbitrary graphs. We demonstrate a sharp threshold on the level of visibility required: at a certain visibility level it is possible to design randomized algorithms that nearly match the best approximation possible even with full access to the graph structure, but with any less information it is impossible to achieve a non- trivial approximation. 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.