Following successful widespread non-pharmaceutical interventions aiming to control COVID-19, many jurisdictions are moving towards reopening economies and borders. Given that little immunity has developed in most populations, re-establishing higher contact rates within and between populations carries substantial risks. Using a Bayesian epidemiological model, we estimate the leeway to reopen in a range of national and regional jurisdictions that have experienced different COVID-19 epidemics. We estimate the risks associated with different levels of reopening and the likely burden of new cases due to introductions from other jurisdictions. We find widely varying leeway to reopen, high risks of exceeding past peak sizes, and high possible burdens per introduced case per week, up to hundreds in some jurisdictions. We recommend a cautious approach to reopening economies and borders, coupled with strong monitoring for changes in transmission.