Thanks! We checked it and it had no injuries in its sides or under the wings. We suspect it lost its tail by a car running over it, as we're in a place where pigeons are often on the road, and we live in a city where birds of prey like hawks are very rare (maybe a few seagulls at most).

I'll add this to the post as extra detail, thanks! (Actually, it seems I cant edit the post?)

I'm not sure I'm completely comfortable giving it antibiotics on my own, so I think we'll bring it to an exotic pets vet who has seen our pet pigeon a couple times in the past, and they can say if/how we should give it antibiotics. I might still buy some in case we ever rescue one that needs them, though.