I am sure that measuring the angle compared with distant stars is the best approach, but you can in fact work out the angle even without looking at distant stars, if you assume the Earth is spherical and you know its radius, and you know both the length and the direction of the great circle between you & your friend.
Here's a terrible drawing:
So here, the radius of the Earth is $R$, and the great-circle distance between you and your friend is $l$. So the angle between you and your friend is $\theta = l/R$ (radians).
Now, you need to agree both to measure the Moon's angle above the horizon at a particularly good moment: that moment is when it crosses the plane defined by you, your friend, and the centre of the Earth. In other words, you need to measure its angle above the horizon as it crosses the great circle between you. This is annoying, because you have to know the direction of the great circle, but it's also nice because you don't need a clock.
So, when you measure its angle above the horizon you get two angles, $\psi$ and $\phi$. And now you have the quadrilateral I drew below the main diagram, and you know three of its angles and two of its sides. The total angle in a quadrilateral is $2\pi$, so this tells you the fourth angle, which is $\alpha = \pi - (\theta + \psi + \phi)$. And you can then go on to solve the quadrilateral for the distance to the Moon, $d$ (I am leaving this out because I can't remember how to do it!)
Note that this will only work if the Moon is above the horizon for both of you when it crosses the great circle: it may not be I think.