An individual electron going around the loop would of course radiate. However, if we have a constant current going around the loop, there are many electrons and their individual radiations cancel each other, so there is no net radiation. They balance each other out. We can show there is no net radiation by solving Maxwell's equations on the net current, which is constant. See, e.g., McDonald's Why Doesn’t a Steady Current Loop Radiate? (pdf).
There would be some transient radiation when you change the current. Also, in the real world, there would be some infinitesimal radiation due to the fact that the current won't truly be constant at the microscopic level, i.e. because of Johnson noise etc.