Management of Distribution Groups synced from on premise AD

[Stephane Favre]

I am trying to get my head around managing of DLs that have been synced from on premise AD but owners that have been migrated to O365 via the hybrid connection. Keep getting permission related errors, and everything I have read eludes to the fact that those DLs need to be managed on prem with exchange management tools.

Is there another way to allow users to mange their own DLs in O365 and to allow changes in O365 to sync back to the on prem exchange environment?

[#2]

I'm sure you already saw this support article:
https://support.microsoft.com/en-us/help/2417592

Other than the suggested options, providing the following shortcut to DG managers might be an option, depending on permissions due to direct group management in the user context instead of via the Exchange Trusted Subsystem:
%SystemRoot%\SYSTEM32\rundll32.exe dsquery,OpenQueryWindow