Physicist: Assuming this question isn’t about interstellar political leanings, the answer is: it’s tricky, but it can be done.
This is worth trying out just once in your life. Try to define left and right for someone using only physical principles, and without making reference to anything else. So you can define “down” by asking the other person to hold something out and then drop it, but you can’t define down as “the direction the Eiffel tower isn’t pointing” (for our Parisian readers). You’ll find that any (successful) attempt you make to distinguish left from right involves citing example.
If an alien is so distant that there are no common landmarks (spacemarks?) that you can both see, then you’re restricted to just using experiments. You can cheat by sending your signal using circularly polarized light, but let’s not cheat.
For a long time it was assumed that the universe can’t tell the difference between right and left, since every physical law seemed to work exactly the same right-ways as left-ways (Also, since the universe has no thumbs it can’t make an “L” with its left hand).
However, the dawning of the nuclear age heralded a new age of handedness. It turns out that if both you and your alien friend have access to decent laboratories, then you can describe left and right using only physical principles.
Some nuclear processes (specifically “β– decay” in what follows) can tell between left and right. Or more accurately, they can tell the difference between right and left-handed “chirality” (this post talks about chirality in passing). So if you get something like radioactive cobalt and stick it into an extremely strong magnetic field you’ll find that the radiation it produces tends to stream out in one direction, which allows you to tell the difference between right-handedness and left-handedness.
One way is to notice that to produce the magnetic field you need to run electrical current in a loop, and the direction of the radiation stream allows you to define whether the current is running clockwise or counter-clockwise (in one case the radiation is toward the ring and in the other it points away). Once you’ve defined clockwise and counter-clockwise, left and right is easy.
There is a caveat. If the same experiment is done on anti-matter, then the results are reversed. So, say you both run the experiment and find that the radiation comes out of the top of both of your devices. While you both think that you’ve managed to communicate left/rightness, you’ll find that you’re mistaken.
But it won’t be until you invite your alien friend to Left-hander’s day that you’ll discover the mistake. Hopefully you’ll manage to catch it before they land their anti-matter spaceship, which would be a bad day all around. Bringing anti-matter and matter together results in the complete annihilation of equal amounts of both, and the release of all of their energy (which is a lot).
Luckily, there doesn’t seem to be any anti-matter floating around the universe. At least, not much. So, if you know that you and the alien are both made of matter (which is pretty certain to be the case), then you can tell the difference between left and right.