Generally speaking, it is the responsibility of the duty-bound (mukallaf) to identify the subject matter. However, it should be noted that identification of certain subjects involves inference and deduction and it would be, therefore, the duty of a Mujtahid to identify it. As well, when the Mujtahid has certain criteria and principles to apply in order to find out the subject, the duty-bound cannot act upon his own view. For instance, when it comes to sighting of the crescent (which has nothing to do with taqlid but with one's certainty), if a Mujtahid does not consider sighting of the moon with equipped eyes as sufficient, the duty-bound cannot consider sighting of the crescent with telescopes and binoculars as sufficient for establishing the beginning of a month.
Indexes:
Answers given by a number of the living grand religious authorities to the same question are as follows:[1]
Grand Ayatollah Khamenei (may Allah grant him long life):
If the subject matter is amongst the religious truths or it is amongst the subjects derived from legal and religious arguments, it is the responsibility of the Mujtahid to identify it, or else the identification of the subject is a duty of the duty-bound (muqallied).[2]
Grand Ayatollah Sistani (may Allah grant him long life):
The cases are different and some of the subjects should be identified by the Mujtahid himself.  
Grand Ayatollah Saafi Gulpaigani (may Allah grant him long life):
Generally speaking, the duty-bound himself should identify and distinguish the subject except for certain cases where the Mujtahid has certain tenets and principles to follow. For instance, a mujtahid does not consider sighting of the crescent with equipped eyes as scientific. In this case, the duty-bound, in this case, cannot consider telescopes and binoculars as sufficient for sighting the moon and therefore ascertaining the beginning of a month.  
Grand Ayatollah Makarem Shirazi (may Allah grant him long life):
In the said case, one can act upon what he says.
Grand Ayatollah Noori Hamedani (may Allah grant him long life):
The subjects are different. It is a responsibility of the jurisprudent to elucidate and clarify the inferred subjects.

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