Large-scale international scientific collaborations are increasingly common in the field of STEM (Science, Technology, Engineering, and Mathematics). However, little is known about the well-being of the members participating in these `big science' collaborations, which can present unique challenges due to the scale of their work. We conducted a survey among members of three large, international collaborations in the field of gravitational-wave astrophysics in the summer of 2021. Our objective was to investigate how career stage, job insecurity and minority status are associated with reported levels of depressive symptoms as well as the desire to leave academia. We found that early-career scientists and certain minoritized groups reported significantly higher levels of depressive symptoms compared to senior members or those who do not consider themselves as a member of minoritized groups. Furthermore, relatively young members, staff scientists/engineers, and those experiencing high levels of job insecurity and lack of recognition were more likely to frequently consider leaving academia. Our findings suggest that improving recognition for personal contributions to collaborative work and providing clearer job perspectives could be two key factors in enhancing the well-being of young scientists and reducing the potential outflow from academia.