People's perceptions about the size of minority groups in social networks can be biased, often showing systematic over- or underestimation. These social perception biases are often attributed to biased cognitive or motivational processes. Here we show that both over- and underestimation of the size of a minority group can emerge solely from structural properties of social networks. Using a generative network model, we show that these biases depend on the level of homophily, its asymmetric nature and on the size of the minority group. Our model predictions correspond well with empirical data from a cross-cultural survey and with numerical calculations from six real-world networks. We also identify circumstances under which individuals can reduce their biases by relying on perceptions of their neighbours. This work advances our understanding of the impact of network structure on social perception biases and offers a quantitative approach for addressing related issues in society.