I'm going to make a guess that you're not using security groups to manage the permissions on these mailboxes. I'd recommend, before you do anything else, that you create a security group for each logical grouping of users who need special permission on each mailbox, place the appropriate users into those groups, and grant permission using the groups.
(Beware-- the users' security token may be cached by the Exchange Server computer and it might not see the update group membership immediately. Depending on your environment it could take up to 2 hours in a stock configuration for the server to "see" the change. More details at http://theessentialexchange.com/blogs/michael/archive/2008/01/18/Exchange-Server-Caches.aspx if you want to know more about what I mean.)
By doing this with a group you eliminate the need to modify the permissions in the future if job roles change, etc, for the people involved. You need only specify the right group memberships for new users, changed job roles, etc, and the permissions "just work".
Using the Exchange Management Shell, grant "Full Mailbox Access", "Receive-As", and "Send-As" permission with the following commands:
Add-MailboxPermission "Mailbox Na,e" -User "DOMAIN\Group NAme" -AccessRights FullAccess
Add-ADPermission "Mailbox Name" -User "DOMAIN\Group Name" -ExtendedRights Receive-As,Send-As
Obviously, substitute in the mailbox name and group name where appropriate.
(Users with Full Mailbox Access don't have "Send As" permission-- see http://technet.microsoft.com/en-us/library/dd421860.aspx for details)
Best Answer
The Mail Recipient Creation role allows an individual to create a recipient. The Mail Recipients role enables a user to manage existing boxes.
It sounds like you want the Mail Recipient Creation role.
Source: http://technet.microsoft.com/en-us/library/dd876911.aspx