I believe the local people usually know where the centre is. They tell you when you ask them. Give it a try, I bet you'll see that you get similar answers.
I would expect this most central spot to be often in front of the townhall (maybe sometimes the cathedral) or the main market square (in traditional towns) or likely both. Maybe the train station can be such a center as well (in more recent towns which developed after the railway was invented).
One criterion could be: where would you go to protest something. If you were to gather together with "all" citizens, where would you go.
Streetsigns aren't useful from my experience. Often the centre is inside the main pedestrian area (typical for Germany, where many historic city centres have been pedestrianized) and it is generally so big that you will get signs in the outskirts indicating "city centre", but when you arrive there will not be a sign stating "this is the city centre", the signs will simply stop indicating a centre when you are in a central area (they might indicate "all directions" now, but that's a different story, maybe to be told another time).
In special cases there could even be several centres, e.g. Berlin (there might be a bit of a difference between big cities and smaller towns). For West-Berliners, the centre was likely Breitscheidplatz (in OSM) and as you can see it is also very close to the former main station of West Berlin, Bahnhof Zoologischer Garten).
For the people in East Berlin it was probably Alexanderplatz (in OSM) with the centre extending also to here (between tower and main town hall. There might be some other opinions because the former centre (castle) was also reclaimed by the officials, naturally (see below). This is Alexanderplatz, you can also see the prominent tower, visible from everywhere in the city. Huge towers are generally issueing a statement of centrality). There's also a station, but it is not important as a train station (it is a very important underground station though, with 3 lines crossing):
Berlin central island
Another central spot and maybe the official centre during the GDR, surely during the German Empire, is the palace (Stadtschloss) (now in reconstruction), (in OSM), in times of the GDR this was the place of the "palace of the republic" (Palast der Republik), at its side there is also the national council.
In this map from 1688 you can see the central island (yellow-green). By that time there were still 2 towns, Berlin (purple) and Cöln (yellow), and this origin is likely a reason that there's some concurrency between Alexanderplatz and the castle. The castle is signed a. Alexanderplatz is in Berlin, while the castle is in Cöln.
The Berlin castle 1900
After tearing down the castle and waiting 20 years (in the meantime they set up a permanent grandstand for their military parades on national holidays), the palace of the republic was errected (after the reunification, the Germans waited again 20 years, until they decided to demolish this "palace" and reconstruct something which imitates the imperial castle's volume and facade).
Tearing down the castle, 1950
Grandstand for parades at the place of the former castle, 1952 (likely not perceived as a centre):
Palast der Republik, 1981:
Closeup of a parade in front of the Palast der Republik, 1978:
Palast der Republik from the inside, 1976
Here's another map to illustrate that this is the centre of the centre (also note how the avenue "Unter den Linden" points straight to the castle (=centre), for many kilometers, and bends right in front of the centre to pass the castle). In red the former castle, gray underneath the palace of the republic, to the north-east Alexanderplatz and its tower):
The current reconstruction called Stadtschloss, (2015). Very visible how Unter den Linden bends for the castle, see it also in a much bigger scale in OSM. This is the main road, leading straight to the castle (current federal road does not completely follow the old road, but I think you can get it) and at the time used by the king to got to his residence in Potsdam. What is now there is in part also result of more recent times: the Nazi plan by Albert Speer).
The OSM crowd currently has set the centre a bit closer to the west, at the crossroads Unter den Linden and Friedrichstraße (in OSM), it's not completely off as this is a very central place (and in the historic parts of Berlin, pre-industrialization), but it isn't a centre where you would go to gather for protest (these would be Pariser Platz in front of Brandenburg Gate and close to the Reichstag, or Alexanderplatz).
I will soon have a look how this node moved around in OSM with the years (and also look for the previous Berlin node that was deleted in 2008).