Generally by it's radiation signature. Black holes don't just swallow matter without a trace. As infalling matter is accelerated to immense speeds, the tidal forces rip it into atoms and internal friction heats it up to very high temperatures.

We'd basically see extremely energetic particles coming from a nearby source. Since no other known physical process can generate such high energy particles (not even fusion in main sequence stars), we'd have to conclude that there is probably some extremely dense object there.