As per Section 25(5) of CGST Act, both institutions at Chennai and Telangana are distinct persons. Students (trainees) are not registered persons. So as per Section 12 (5)(b) of IGST Act place of supply is Chennai. Therefore, CGST & SGST are to be paid.
SECTION 12 of IGST Act. Place of supply of services where location of supplier and recipient is in India
(5) The place of supply of services in relation to training and performance appraisal to, -
(a) a registered person, shall be the location of such person;
(b) a person other than a registered person, shall be the location where the services are actually performed.