Unfortunately this is too long to be a comment on the answer of M. A. Golding.
Impacts to the head, concussion and unconsciousness: As Trioxidane noted in a comment on the question, Hollywood gives a very misleading view of how blows to the head turn out. The first concept to look at is how a concussion is caused. In non-technical terms, it is the brain getting bounced around inside the skull. Due to the spine not compressing very well, this does not happen much as a result of blows from above that strike the top of the head, as PcMan related anecdotally. Impacts from this angle are most likely to have either minimal effect or fracture the skull and cause a serious and possibly fatal brain injury (but not necessarily immediate unconsciousness). (Fortunately PcMan suffered the former and not the latter.)
In order to maximise the chance of bouncing the brain around (ie causing a concussion), the head needs to be accelerated forwards, backwards or to the left or right. This article suggests that impacts from the side are more likely to cause a concussion than impacts from the front or back, although the data at the time was suggestive rather than conclusive. What this means is that if a bird's biowaste projectile is going to concuss rather than either annoy or kill someone, the bird needs to be flying very low and very quickly so that sideways rather than downwards force is delivered to the head.
(Note that even with ideal concussion-producing conditions, only 1 in 10 concussions results in noticeable loss of consciousness.)
The next part is looking at how much force is required and how it is delivered. Googling "speed of MMA punch" throws up a few results - I'm going to cherry pick a conversation in which an unnamed study apparently found 9.14 m/s and round up to 10 m/s. Let's say that the arm throwing the punch masses 4 kg (ignoring the body behind it), so we're looking at kinetic energy of 200 J.
10 m/s is a pretty puny airspeed but 4 kg is a massive amount of bird doo doo. Let's double the speed - giving 4x the kinetic energy, which brings the payload mass down to 1 kg. For a really big bird this is vaguely believable - the ostrich loses only 2.3 L/day of water through evaporation, urine and feces under harsh conditions, but that is enough for 1 kg fecal bombs. Assuming that the fecal matter has the same density of water and is spherical in shape it is roughly the same size as a fist, so there are no concerns about overpenetration.
There it is - an at-least-ostrich-sized bird with a decent horizontal airspeed (72 km/hr) that flies across rather than along paths (if any) to release big droppings. Without vast numbers of really big birds the risk of being struck would be very low, so presumably this is deliberate behaviour used to hunt or drive off intruders in their territory. There would need to be a reason for evolution to favour horizontal bombing runs over the more easily aimed dive bombing technique. Perhaps it is used to strike intruders who have taken cover under trees or rock overhangs.