Apparently by default, when you go into a Distribution Group and go to the Group Information Tab and assign users to the Managed by: area, this is not enough to allow them to make changes. After some investigating I found the correct PowerShell commands to run to allow them to modify groups they are managing, and not allow them to create or delete distribution groups.
New-ManagementRole -Name OwnerDistributionGroups -Parent MyDistributionGroups
Remove-ManagementRoleEntry OwnerDistributionGroupsNew-DistributionGroup -Confirm:$false
Remove-ManagementRoleEntry OwnerDistributionGroupsRemove-DistributionGroup -Confirm:$false
New-ManagementRoleAssignment -Role OwnerDistributionGroups -Policy “Default Role Assignment Policy”
Our trainer was going to sit down with the users that were managers of these groups. She requested a list of the distribution lists, along with who was able to manager the lists. Ran the following command to get the ManagedBy info:
Get-DistributionGroup | fl Name,ManagedBy > C:distro-managedby.txt