When attempting to insert a row on a m2m list inside of an embedded list as a non admin, there is a good chance, you will receive a security constrain error.
The error will say, "Security Prevents Writing to this Field" below the search field.
The reason this is happening is that there is a global (*.none) ACL for list edit which has a role of Admin.
This will restrict the use from editing the list is they are not an admin which is what they are doing in an embedded list adding a row.
Create a new list edit ACL which is for the table that is in question that follows the best practices of your business requirement.