We study the transmission of monetary shocks across euro-area countries using a dynamic factor model and high-frequency identification. We develop a methodology to assess the degree of heterogeneity, which we find to be low in financial variables and output, but significant in consumption, consumer prices, and variables related to local housing and labor markets. Building a small open economy model featuring a housing sector and calibrating it to Spain, we show that varying the share of adjustable-rate mortgages and loan-to-value ratios explains up to one-third of the cross-country heterogeneity in the responses of output and private consumption.