No, they will roll them later. We track how many turns since they went down. As soon as someone checks on them (successful DC 10 Wisdom (Medicine) check) we roll. Sometimes this is during the battle still, other times it is after it is over.
Once the person is successfully checked on, they make as many death saves as round had passed. This might lead to a sigh of relief when the character is stable, or pangs of regret if they have died (Why, oh mighty Zeus?! WHY!?! If only we had gotten to him in time!).
The idea is players will react differently with the knowledge that a fallen comrade is making saves and stabilizing or failing and dying. But how would they know?? They can't tell and it shouldn't be player knowledge, either. This way other characters will spend resources or at least actions to act when a comrade is down instead of "oh, he has made two death saves already? well, I don't need to bother."