Unfortunately, no.
Usable: You can use Private Email only by your domain name. i.e. sales @ yourdomain.com / sales @ yourdomain.info / info @ yourdomain.net
Unusable: Like, you @ gmail [.] com / you @ yahoo [.] com / you @ anything [.] ltd. And, if you set such other than your authentic domain email, it will not work.
To learn more... How do I get Private Email for Reseller Store?
To learn more... Why are my Store Users getting emails from IMEI Source email account?