To change permissions for a distribution list, login to the MSPControl and click on Distribution Lists,
or go to Organization > Exchange > Distribution Lists.
You will see all Distribution Lists. If you have many, it might be a good idea to use Search form in the right to find the one you need. You can choose whether you want to search by Display Name or E-mail Address.
Once you found the list you were looking for, click on its display name.
You will see settings page with five tabs: General, E-Mail Addresses, Mail Flow Settings, Permissions, and Member Of. Go to Permissions tab.
You can change settings for On Behalf Of, and also Send As settings for the list.
Send on behalf means that selected users can send messages on behalf of that list, and his correspondents will be able to see who sent this message on behalf of the list. E.g. John Doe on behalf of sales_eu@customer01.com.
Send as means the correspondent will see that the message was sent by sales_eu@customer01.com, and will not see the exact user who sent it.
Select mailboxes from the drop-down list and click Add.
The same procedure applies when you want to grant the permission to Send As to certain users.
Don't forget to click Save Changes!