I believe parishioners should open their mouths once during a Traditional Mass - when receiving the communion host. The rest of the time, mouths should be shut.
Two issues jump out:
1. When serving on the altar, it is a distraction for me to hear non-server responses. For example, members of the congregation will respond 'Deo Gratias' or 'Laus tibi Christe' before the servers.
2. In some chapels with a choir, parishioners will sometimes chant responses. This leads to problems as the choir will raise voices to a higher level in an attempt to drown out the parishioners. The choir volunteers spend endless hours to prepare for High Mass, and listening to off-key parishioners spoiling the grace and beauty of the choir's chant is disturbing.
Some SSPX pastors agree with the above, but lately some are encouraging sing along.