Great danes can drool, just like any other dog can drool. It depends on the shape of the dogs lips on how much the dog will drool. A lot of the European lines have dogs that have a lot looser lips so they tend to drool more than a lot of the americanized danes.
There is no "dry mouth" bred dane. Some only drool when drinking water or are getting a treat, some only when they are nervous and some all the time. Lip size as mentioned above does play a role but not always. We share our home with 7 danes. 3 are american, 1 is 1/2 american and 1/2 european and 3 are european......in our home my smallest lip girl is drooling all the time where my biggest lipped male only drools when drinking. The others are a mixture. We have one girl that drools only when she is nervous. So yes danes do drool....some to varying extents but its not something that you can know when they are babies or from there parents.
Large dogs usually drool a lot especially if they think they are going to get a bite to eat.