Any hydraulic failure constitues a Full Emergency so it is essential to advise ATC.
If it is being classed as a full emergency, then I guess it depends on who is doing the declaring.
For a pilot in an aircraft with triple hydraulic systems, the loss of one might only increase risk margins slightly, but is certainly not going to cause the aircraft any handling problems. You might not even need to declare a PAN.
The ATC definition of a full emergency, for aerodrome emergency orders, is one where the aircraft is known or suspected to be in such trouble that there is the danger of an accident.
Where the pilot has advised of a hydraulic failure but is able to confirm to ATC that the defect will not normally involve any serious difficulty in effecting a safe landing, then a Local Standby fits the bill.
Provided you communicate the latter, there should be no need for ATC to do anything over and above that they would for any other local standby.
It's all down to communication. Be clear about the problem and it's effects on the operation of the aircraft. Then ATC can react accordingly
