Sending email from different domain (a company we are merging with)

We are currently in the process of merging with a company. We are still in discussions whether we are going to create an Active Directory Trust.

In the meantime we have both federated our exchange environments with Microsoft. So we can now see each others free/busy information. They have 2010 on prem, we have 2016 on prem.

Is it possible for people to send emails from domainA having domainB's domain name in their smtp address ?

Would this be any easier if er did have an active directory trust in place ?

Thanks 

db