I have always understood that a loose airside dog will be shot - risk of getting onto a runway, causing a swerve / go around / collision etc. Possibly told by the Tarmac Terriers at some stage, but don't recall seeing anything on a sign.
It was a minor problem for me because the boss's wife would bring her bucking fig Rottweiler with her. She would hold it on the lead, but it easily got away from her a few times. Got a nice scar on my hand from when I first met it. ("Hello doggie!" Chomp! "Don't try to touch the dog." (could have said that before I tried, mumble mumble...)) It had its own boarding ramp and piddle-proof mat that had to travel with it. But it still found ways of piddling on the carpet, and the engineers were always HAPPY to pull the carpet out and wash it...