as a controller, i weigh a number of factors into the equation...
for example, if i am speaking with a GA pilot, i will not hesitate to declare the emergency for him/her if i suspect the pilot is hesitant to do so or if i suspect that something is happening that he/she is either not disclosing right away or underestimating the gravity of a situation developing such as icing.
from my end, i listen to the sound of the voice (confusion, fear, discomfort, hesitancy, lag in response time in comm.) as well as the behaviour of the radar target (ground speed variations, directional variations, inability to maintain steady altitude) . i do not necessarily always advise him/her of what i am doing in declaring an emergency on his/her behalf as there is no need to do so. i will however give priority to the aircraft which may mean clearing other aircraft off my frequency, vectoring traffic around him/her, and not allowing other traffic onto my frequency to limit congestion, so that i can be more available to that person in particular. coordination-wise, i make all adjacent sectors that may be affected aware of the situation in the event that things should turn for the worse. geographically, i locate all the airports within close proximity to the a/c, access all pertinent airport information for the pilot in case it's needed, look at the weather and the winds at the airport so that i know which runway at any particular airport i'll need at a moment's notice. i take whatever help i can get from fellow controllers nearby in assisting me with accessing and coordinating what i may or may not need. if weather is a factor, i query any nearby commercial pilots to pan out their weather radar to assist as their weather radar is much more reliable and accurate than mine. operationally, i become much more aware of every aircraft nearby, on or off my frequency that can assist if i need help ( i have in the past sent aircraft to locate another aircraft that went down so that emergency rescue could locate the down aircraft more easily), i have other controllers on standby that will assist with information accessing and coordination, and i locate any controllers who are pilots to be ready to assist with cockpit management/control if need be. since controllers work a defined area of airspace, other controllers in adjacent sectors will move to assist in my area so that the pilot never has to change frequencies.
for a commercial pilot, the same applies for the most part. the cockpit is hectic so conversations are usually limited to what's necessary until i know the PIC has the aircraft under control. things usually happen at a much quicker pace for the most part with commercial carriers as they usually are much more hesitant to declare an emergency. so a commercial carrier declaring an emergency usually puts a controller right down the crapper when time is of the essence.
in the end, any paperwork to do is worth a life. i ultimately have to go to bed at night and sleep in peace. i'd rather go down the crapper erring on the cautious side than not ever be able to sleep at all because i didn't.
i get paid to keep planes apart but i also get paid to do whatever i can to help any pilot that sounds like they need help, especially in an emergency.
whether it's a mayday or a panpan, our actions as controllers remain the same for the most part.