Email Notification for User

Will this field be changed only after the record has been created? If so this will work for you.

Create a new notification with the "Notify Whom" set to the [Assigned To] field(NOT the one that only goes to you) and have the criteria be when a record is modified. Click additional criteria and then "any of the following fields change" and select [Assigned To] then in the filters you can do something like "[Assigned To] does not equal Blank" and whatever other filters you need. This one will work assuming all you care about is the field being filled in and that it gets filled in later after the record is created,