The answer depends on the intensity of the beam and its energy. As L.Dutch's answer has correctly note, each of the positrons in the beam will eventually annihilate an electron, emitting a pair of gamma rays. But that not the most important damage it will do.
The main mode of damage happens because the positrons in the beam are charged particles which strongly interact electromagnetically with the electrons in the air, first dumping their kinetic energy into them and then annihilating them. So a positron beam in air will ionize the air, forming a column of plasma the length of the beam. Since the plasma interacts even more strongly and you, basically, get a lightening bolt. (Note: this is without the annihilation bit happening at all! An electron beam will worm pretty much the same way. This is why electron-beam welding is done in a vacuum.) See this for some more discussion.
If you make the beam less energetic so there's less kinetic energy to be dumped into the air, the beam just annihilates in a shorter distance. If you make it intense enough to just cut a hole right through the air, you turn your lightening bolt into a coruscating beam of ravening energy(1).
Shooting an electron beam is destructive enough. Shooting positrons adds to the fun by annihilating electrons, too, but this does surprisingly little additional damage compared with the effect of the beam's electromagnetic interaction. The energy of annihilation goes entirely into a pair of gamma rays. The gamma rays are mostly absorbed by nearby matter, but the damage they do is small compared with the damage done by the charged particle beam itself. (The gamma rays produced are 500 KeV gammas, and they dump their energy over tens of centimeters of flesh. The incoming positrons have more than 1 MeV of kinetic energy and dump that energy into a narrow line through the body where the beam passes. The effects are much more devastating. ...Which is not to say that the gamma is harmless, it's just that you're so dead from the energy dumped by the electromagnetic interactions you don't care.)
(It's worth noting that a beam of positrons is not stable for two reasons. First, since positrons, all being positively charged, repel one another. No matter how narrow the bean is when it is created, it will widen. If the positrons are fast-moving, though, the beam can get further before falling apart. (This is why the particle beam weapons projects that survived the longest were neutral beam weapons.) Secondly, interactions with the plasma it creates increases the instabilities.)
So the bottom line is that if you have a sufficiently intense beam of positrons to get to the target, you're shooting something like a lightening bolt and (in air) you can't avoid the damage due to the purely electromagnetic effects being far greater than that due to the electron annihilation.
Note the "in air" part: In a vacuum, it's a different matter entirely and the physics of the beam are quite different.
(1): See the works of E. E. Smith, PhD.