Inadequate COVID-19 testing capabilities is producing testing data that cannot be reliably compared across countries or across jurisdictions within low- and middle-income countries (LMICs). This is hampering the ability of LMICs to devise timely and effective policy responses, such as identifying hotspots and spatially targeting public health responses or economic relief. We develop a methodology to make indirect inferences about the spatial distribution of COVID-19 risk using the insight that migration is a primary driver of disease spread across jurisdictional borders. We use bilateral migration flows and prevalence of COVID-19 cases in all migration destinations (such as Italy, Spain, United States, Singapore, and South Korea) to construct a country-level index of COVID-19 risk exposure. We then use sub-district and municipality level data on the origins of migrants, and airport disembarkation data on recent returnees, to create sub-national heat maps of COVID-19 risk within Bangladesh and the Philippines. We validate our indices by comparing them to data on confirmed cases, COVID-19 deaths, location of quarantines and distress calls to a government hotline. We use multiple data sources in Bangladesh to evaluate the broad applicability of our method to other LMICs where only a subset of such data may be publicly accessible.