Say your conference room mailboxes are not processing meeting requests from Office 365 or from partner organization, and you want to process those meeting requests too, you come to right place.
Two ways you can accomplish that:
One of Two: On the Exchange Connector that receives emails from Office 365 or from different mail servers, Enable Externally secured (for example, with IPsec) in Security Tab of the connector properties.
Two of the Two: If you don’t like the above idea, you can enable ProcessExternalMeetingMessages setting on your conference room calendar processing settings. Whip out an Exchange Shell, the type this for every conference room.
Set-CalendarProcessing -Identity <Conference Room Name or Email Address> -ProcessExternalMeetingMessages:$true
Anand, the Architect 
HI,
I”m having this issue and i’ve currently enabled the processexternalmeetingmessages to true. I’m in a O365 hybrid enviornment with on-prem Exchange 2010. If I use the ‘externally secured’ in my on-prem environment, will it work with Exchange 2010 and is there anything I need to confirm is configured in O365 for it to work?
Yes. It will work.