The ability to bypass Google login restrictions is a policy level setting. This means that like any other policy setting, it also requires users to first authenticate themselves before the policy is applied.
As such, teachers would need to first log in using their school Gmail to allow the system to recognize them and determine if personal Gmail restrictions are to be bypassed. Once logged in, they can then log in to their personal Gmail account.
To learn more about restricting personal Gmail access, click here.