Online social networks are ubiquitous parts of modern societies and the discussions that take place in these networks impact people's opinions on diverse topics, such as politics or vaccination. One of the most popular models to formally describe this opinion formation process is the Friedkin--Johnsen (FJ) model, which allows to define measures, such as the polarization and the disagreement of a network. Recently, Xu, Bao and Zhang (WebConf'21) showed that all opinions and relevant measures in the FJ model can be approximated in near-linear time. However, their algorithm requires the entire network and the opinions of all nodes as input. Given the sheer size of online social networks and increasing data-access limitations, obtaining the entirety of this data might, however, be unrealistic in practice. In this paper, we show that node opinions and all relevant measures, like polarization and disagreement, can be efficiently approximated in time that is sublinear in the size of the network. Particularly, our algorithms only require query-access to the network and do not have to preprocess the graph. Furthermore, we use a connection between FJ opinion dynamics and personalized PageRank, and show that in $d$-regular graphs, we can deterministically approximate each node's opinion by only looking at a constant-size neighborhood, independently of the network size. We also experimentally validate that our estimation algorithms perform well in practice.