Hi Cary,
Atricle suggested by Allen, should definately get the things done smoothly.
However you might be intersted to learn\refresh about "accepted domains, Authoritative and Internal Relay"
It should answer your question: "This should have some conflicts because duplicate email address and their domain was registered in our DNS server."
Basically, No there will not be any conflict for DNS domains. it will be handled by the Exchange Servers. Your public DNS points to single location either On-prem or O365 (On-prem in your case is better, as most of the users are here)
O365 way:
Internal relay Selecting this option means that recipients for this domain can be in Office 365 or your own email servers. Email is delivered to known recipients in Office 365 or is relayed to your own email server if the recipients arent
known to Office 365.
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You should not select this option if all of the recipients for this domain are in Office 365.
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If you select this option, you must create an outbound connector; otherwise recipients on the domain who are not hosted in Office 365 wont be able to receive mail on your own email servers.
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This option is required if you enable the subdomain routing option on a domain in order to let email pass through the service and be delivered to any subdomains of your accepted domains.
On-Premises Way:
Internal Relay
Domain E-mail is relayed to an e-mail server in another Active Directory forest in the organization.
An organization may have to share the same SMTP address space between two or more different messaging systems. For example, you may have to share the SMTP address space between Exchange and a third-party messaging system, or between Exchange environments
that are configured in different Active Directory forests. In these scenarios, users in each email system have the same domain suffix as part of their email addresses.
To support these scenarios, you need to create an accepted domain that's configured as an
internal relay domain. You also need to add a Send connector that's sourced on a Mailbox server and configured to send email to the shared address space.