If I am interested in mechanical engineering, which is better, COEP, the College of Engineering Pune, or VJTI in Mumbai, and are they both comparable to VNIT in Nagpur?

As a student you should check the following areas and take them into consideration while choosing a particular institution. 

Research- VNIT, and COEP beats VJTI in these criteria. Students can do research here under college professors on winter vacations.

Placements-

• VNIT provides unpaid internships to students through the alumni association.
• IT companies offer job opportunities to people from computer science, electronics, electrical with a CGPA of 7. The same scenario is present in VJTI and COEP. They may provide better opportunities than VNIT.
• However, there are decent placements for mechanical engineering both in VJTI and VNIT.

Campus Life-

• VNIT is best in terms of campus life. There are ample hostel facilities for the students.
• Students can interact and socialize with people not only from Maharashtra but from all over India.
• There are almost eight batches for the department with 60-90 students.

Student Activities- VNIT excels in expectations than COEP & VJTI with myriad festivals, clubs, and extra-curricular.

Research after Bachelors’- Lots of VNIT students get themselves into research work after their graduation. However, there is no available information for VJTI and COEP.

Considering all the aspects listed above, VNIT seems a better choice for mechanical engineering than COEP Pune and VJTI Mumbai.