It’s not always so clear about that. In Passaic, when the community was forming the local Hatzalah, R’ Meir Stern (the Rosh Yeshiva in Passaic) was against it. He felt the local 911 system was fine, and Hatzalah was not needed, and would cause unnecessary chilul Shabbos. They went ahead and founded it over his objections anyway.
In Teaneck, there is no Hatzalah. There is a local volunteer EMS that is mostly Jews anyway, and their response time is excellent. On Shabbos, it is staffed by non-Jews, but the response time is still great.