I don't mind that the faithful do some of the responses at High Mass at the SSPX chapel, and it's always been done that way (well, for quite awhile), but I wouldn't call it specifically a Dialogue Mass. It's not an intentional Dialogue Mass.
It's quite different from the local FSSP Mass, where the faithful tend to be very quiet, even in singing the processional and recessional hymns. At least at the SSPX chapel, the faithful sing along, and mostly they don't even need a hymnal (though I do).