Different full moon face on different places on Earth same time
Yes, they do see a slightly different face of the Moon.

It's easy to calculate how different. If they're on nearly opposite sides of the Earth so that one sees the Moon just rising and the other sees the Moon just setting, they are seeing it from about 1.9 degrees apart. (A triangle with a base of 8000 miles (the distance they are apart) with sides of 240,000 miles (the distance to the Moon) yields an angle of 1.9 degrees (8/240 radians).

So they'll be viewing the Moon from 1.9 degrees apart and will see a full moon displaced by 1.9 degrees, or about 36 miles on the surface. So one will see 36 miles of Moon more on the Lunar east side than the  other and the other will see about 32 miles more on the Lunar west side.