Using coolant, not overheating and not leaking

I just bought a car and I took it for a test drive. I was only gone for ten minutes and when I got back I noticed what looked like steam coming out from under the hood where the radiator is. I mentioned it to the seller and he said that he must have forgot to add water to the reservoir when he changed the thermostat. We added water in the reservoir and the water drained within a couple of minutes with no leaks under the car. We added more water and while the car ran and eventually the water remained at a consistent level. When I got the car home I went for a drive and could smell radiator fluid. So I checked the coolant level and it was empty. Once again I filled it and went through the same thing that happened when I bought it. Once it held water I drove it home an checked the level and its empty again. The car runs really nice. The temperature gauge doesn't indicate that it's overheating. I haven't seen anything come out of the exhaust. It starts without a problem and runs quietly. What is your best guess to what could be wrong with it. Can I continue to drive it short distances without possibly doing more harm?