Can a pilot (other than an ATP) who is not certificated and current as a CFI give instruction in aircraft required for the issuance of a pilot certificate, e.g. a private pilot certificate?
I'm assuming that you're asking about FAA regulations. Anyone can train anyone else - how effective and safe that training is is a different question! - but only an instructor can issue endorsements, including signing off on training time, endorsing solo flight, and accepting a student pilot application (see 14 CFR 61.193 and 61.215).
This is from 14 CFR 61.51:
(h) Logging training time. (1) A person may log training time when that person receives training from an authorized instructor in an aircraft, flight simulator, or flight training device.
(2) The training time must be logged in a logbook and must:
(i) Be endorsed in a legible manner by the authorized instructor; and
(ii) Include a description of the training given, the length of the training lesson, and the authorized instructor's signature, certificate number, and certificate expiration date.
In theory, a non-CFI could train a student to perform as required by the ACS. But, a non-CFI couldn't authorize the student to fly solo, and he couldn't endorse the student's training time in his logbook. The student would end up able to fly but unable to get a certificate because he'd have no official training time and no solo time. So it wouldn't really be helping the student much: he'd have to go and do the required hours over again with a CFI anyway.
A person who holds a Ground Instructor certificate can. Persons who hold a Basic Ground Instructor certificate may provide ground instruction, knowledge test endorsements, and ground portions of a flight review required for Sport, Recreational, or Private Pilot certificates. Persons who hold Advanced Ground Instructor certificates can perform ground training and the required endorsements to take a knowledge tests for any airman certificate or rating, except instrument ratings. They may also perform and sign off any airman for the ground portion of a flight review. Instrument Ground Instructors may provide ground training, knowledge test recommendations and training and sign off for the ground portions of an instrument proficiency check. The person who only holds a Ground Instructor certificate cannot, however, provide any flight instruction, including instruction in a ground based flight simulator or flight training device.
Airmen who hold an ATPL may give ground and flight instruction, but this is limited to air transport operations as prescribed in §61.167, unless said individual also holds as Flight Instructor certificate.