I am looking for the DOD standard responsibilities/requirements (if there is/are any) of an Assistant Government Technical Representative (AGTR).
This response is based on the information provided. We suggest you discuss with your contracting officer/contract administrator as appropriate.
DoDI 5000.72 dated March 26, 2015, DoD Standard for Contracting Officer’s Representative (COR) Certification, should provide you some of the information you are asking about.
A Contracting Officer's Representative (COR) or Contracting Officer's Technical Representative (COTR) has duties specifically aligned to the assigned contract. The government has training for a COR.
The duties of a COR are related to the Quality Assurance Surveillance Plan, referred to as the QASP. The assignment and specified duties of the COR are delineated in the contracting officer's Letter of Delegation.