What would be the better choice - ECE at BITS Goa or ECE at NIT Trichy?

Since you have decided on a branch beforehand, comparing two institutes becomes easier. That being said, BITS Goa is a clear winner and you can go for it without any hesitation. Consider the following parameters

Infrastructure: Being a private institute, BITS Goa has excellent infrastructure and facilities. The campus is fully wifi-enabled with air-conditioned classrooms, well-equipped labs, and a well-stocked e-library. There are several MOOC or reverse classroom courses too. 

Faculty: BITS Goa has a thorough recruiting process and only then best professors are selected thus maintaining the standards of teaching here.  The faculty members are expected to be highly experienced and have published at least 5 papers. There is also a faculty monitoring committee, to ensure quality teaching.

Course structure: BITS Goa has a well-structured curriculum and the pedagogy adopted is modern.  

Campus life: Being located in a tourist city, the campus life experience is full of fun.

Placements: BITS Goa placements statistics are excellent and this year the average package stood at INR 30.37 LPA.

Unless you have any financial constraints, choosing BITS Pilani over NIT Trichy will prove to be a more fruitful decision.