jportzer and others; ATC declares emergencies all the time without calls from pilots. Often they are for missed position reports and the like which are later resolved but more often at airports where Tower sees something is wrong and hits the crash alarm. By the way these are not declared emergencies involving rescue coordination centres,they are airport emergency procedures.You can always send the firies back but there is no excuse for them being late at the crash site.
Likewise if you see an airplane in the circuit trailing smoke but not talking, or an A320 that stops climbing and starts (and keeps) descending after take off without calling. An airport emergency will be declared, no-one is waiting for a Mayday or Pan call to be the trigger though it does help with information. Towers worldwide typically have the ability to access the public emergency system, ie, Police Fire Ambulance normally without an intermediary.
By the way this would not occur if the aircraft is out of sight and pilot is talking but not making it clear what his problem might be; that's when you might hear the question "Are you declaring an emergency?"