Python 3, 74 bytes

def f(x,y):
 i=1
 while i*10<10**len(x or y)or str(i)in x+y:i*=2
 print(i)

Prints lowest integer with the same length as the first of the inputs that has nonzero length.

Python 3, 74 73 bytes

-1 byte thanks to Step Hen

def f(x,y,i=1):
 while i*10<10**len(x or y)or str(i)in x+y:i*=2
 print(i)

Prints lowest integer with the same length as the first of the inputs that has nonzero length.