If we take a typical string such as this one made by Ningbo Laihe Christmas Gifts Co.,Ltd, the power supply is 30V (for isolation and so it can use low voltage wire). The total power is only 9W. That's about the power used by just two C6 15V incandescent lamps (series string) that were common when I was a child.
The 1000 lamps could be arranged into series strings of 8 LEDs with a single resistor (perhaps molded into the base or cord) in series with each string. There could be a resistor in series with each lamp of 1/8 the value, but that would be wasteful.
Then put 125 of those series strings in parallel to operate from the 30V source. The 30V would be produced by a switching power supply adapter similar to a phone charger or other modern mains-operated switching supply.