If you are planning to use the process builder/flow that invokes the MassMailer actions, you need to assign the MassMailer Apex permission set to the non-MassMailer users.

You can do this by going to MassMailer Setup -> Users -> Click on 'Assign Permission to Non-MassMailer Users'

Search for the respective users to who you want to assign the permission set, select those users and assign the permission set.