Who can perform different maintenance checks?
Airlines are free to conduct the tests wherever they prefer as long as they meet the required FAA standards.

An extract from American Airlines' Procedures:

"PS" Daily Checks

Each aircraft is checked every 2-3 days in its "PS" (Periodic Service) check. The aircraft is visually inspected and its maintenance log book is checked for entries and maintenance needs. The "PS" check can be performed overnight or during downtime during the flight day. It averages approximately two manhours.



"A" Checks

The "A" check is more detailed than the "PS" check. "A" checks are performed every 7- 9 days (approximately 80 - 100 flight hours). The "A" check is performed at one of 40 stations around American’s system. It averages 10 - 20 man-hours.



"B" Checks

The "B" check is an even more thorough maintenance check. The "B" check is accomplished approximately every two months (roughly 500 - 600 flight hours). Besides specific service performed on the aircraft, a detailed series of systems and operational checks are performed. American always performs "B" checks inside one of its hangars at seven different cities around its system. A "B" check requires approximately 100 man-hours on narrowbody aircraft (those with only one aisle) and approximately 200 - 300 man-hours on widebody aircraft (those with two aisles).



"C" Checks

The "C" check is the most thorough type of maintenance work performed by American. The airframe - virtually the entire aircraft - goes through an exhaustive series of checks, inspections and overhaul work. It is performed at either of American’s heavy maintenance and engineering centers in Tulsa, Oklahoma or the Alliance Maintenance Facility in Fort Worth, Texas. There are different levels of "C" checks depending on the type of aircraft.



Jet Engine Overhauls

Modern jet engines are among the most reliable devices in aviation. American does not replace and overhaul jet engines at a specific number of hours. Instead, American uses a 24- hour-a-day "condition monitoring" process that scientifically tracks the condition of every engine on every aircraft. Besides visual inspection, technicians monitor the internal condition of every engine, using such procedures as boroscope inspections and oil sample spectographs. The goal is to replace and overhaul an engine before a problem can occur. Engine overhauls are performed at the Tulsa and Alliance-Fort Worth Maintenance and Engineering facilities. The engine replacement is usually performed at one of the six "B" check hangar locations around the country.



To answer your question on the qualifications required to perform these checks, we'd have to go to FARs Part 65 which lists the privileges and limitations of a Mechanic Certificate.

In addition to the basic Mechanic Certificate, FAA allows two different ratings.

The first one is the Airframe Rating which, under §65.85, states:


  A mechanic who holds an airframe rating may approve and return to
  service an airframe, an appliance, or any related part after he or she
  has performed, supervised, or inspected minor repairs or alterations.
  He or she may also perform the maintenance actions required for a
  major repair or alteration, and should initiate the appropriate form
  (FAA Form 337, Major Repair and Alteration) associated with that work.
  However, the return to service action must be accomplished by a
  certificated A &amp; P technician holding an Inspection Authorization
  (IA). (Refer to 14 CFR §65.95.)


After that comes the Powerplant Rating under §65.87 which states:


  Similarly, a mechanic holding a powerplant rating has the same
  limitations imposed regarding the powerplant and propeller as the
  airframe technician has on the airframe rating. He or she may perform
  and return to service minor repairs or alterations. He or she may also
  accomplish the work activities required for a major repair or
  alteration, but the work must be signed off for return to service by
  an IA.


That said, after all the checks have been performed, a person with Inspection Authorization (under §65.91) is responsible to approve the aircraft for return to service.

Source: American Airlines Aircraft Maintenance Procedures Fact Sheet