The letter leaves your box and never lands in the other firm’s box. A common cause is an empty list of who may send mail for the name. That list is SPF. It is a TXT record.
The record names the servers the mail service gave you. If you send from the hosting mailbox and from a separate business mailbox, both belong on the line. Deleting one drops the other’s letters too. You read the record on the DNS screen. The mail card is on the mail shelf.